Filling the gap between Turán's theorem and Pósa's conjecture
aa r X i v : . [ m a t h . C O ] N ov FILLING THE GAP BETWEEN TUR ´AN’S THEOREMAND P ´OSA’S CONJECTURE
PETER ALLEN*, JULIA B ¨OTTCHER † , AND JAN HLADK ´Y ‡ Abstract.
Much of extremal graph theory has concentrated ei-ther on finding very small subgraphs of a large graph (Tur´an-type results) or on finding spanning subgraphs (Dirac-type results).In this paper we are interested in finding intermediate-sized sub-graphs. We investigate minimum degree conditions under whicha graph G contains squared paths and squared cycles of arbitraryspecified lengths. We determine precise thresholds, assuming thatthe order of G is large. This extends results of Fan and Kier-stead [J. Combin. Theory Ser. B 63 (1995), 55–64] and of Koml´os,Sark¨ozy, and Szemer´edi [Random Structures Algorithms 9 (1996),193–211] concerning the containment of a spanning squared pathand a spanning squared cycle, respectively. Our results show thatsuch minimum degree conditions constitute not merely an interpo-lation between the corresponding Tur´an-type and Dirac-type re-sults, but exhibit other interesting phenomena. Introduction
One of the main programmes of extremal graph theory is the study ofconditions on the vertex degrees of a host graph G under which a targetgraph H appears as a subgraph of G (which we denote by H ⊆ G ).Tur´an’s theorem [21] is a prominent example for results of this type.It asserts that an average degree d ( G ) > r − r − n forces the copy of acomplete graph K r in G (and that this is best possible), where hereand throughout n is the number of vertices in the host graph G . Moregenerally, the celebrated theorem of Erd˝os and Stone [5] implies thatfor a fixed graph H the chromatic number χ ( H ) of H determines theaverage degree that is necessary to guarantee a copy of H : If H has Date : January 6, 2019.* DIMAP and Mathematics Institute, University of Warwick, Coventry, CV47AL, United Kingdom.
E-mail : [email protected] . † Zentrum Mathematik, Technische Universit¨at M¨unchen, Boltzmannstraße 3,D-85747 Garching bei M¨unchen, Germany.
E-mail : [email protected] . ‡ Department of Applied Mathematics, Faculty of Mathematics and Physics,Charles University, Malostransk´e n´amˇest´ı 25, 118 00, Prague, Czech Republic andDIMAP and Department of Computer Science, University of Warwick, Coventry,CV4 7AL, United Kingdom.
E-mail : [email protected] .PA was partially supported by DIMAP, EPSRC award EP/D063191/1, JB byDFG grant TA 309/2-1, JH by the Charles University grant GAUK 202-10/258009,by DAAD, by BAYHOST, and by DIMAP, EPSRC award EP/D063191/1. chromatic number χ ( H ) = r and d ( G ) ≥ ( r − r − + o (1)) n , then H is asubgraph of G . This settles the problem for fixed target graphs (withchromatic number at least 3), that is, graphs that are ‘small’ comparedto the host graph.Dirac’s theorem [4], another classical result from the area, considerstarget graphs that are of the same order as the host graph, i.e., so-called spanning target graphs. Clearly, any average degree conditionon the host graph that enforces a connected spanning subgraph mustbe trivial, and hence the average degree needs a suitable replacementin this setting. Here, the minimum degree is a natural candidate, andindeed, Dirac’s theorem asserts that every graph G with minimumdegree δ ( G ) > n has a Hamilton cycle. This implies in particularthat G has a matching covering 2 ⌊ n/ ⌋ vertices.A 3-chromatic version of this matching result follows from a theo-rem by Corr´adi and Hajnal [3]: the minimum degree condition δ ( G ) ≥ ⌊ n/ ⌋ implies the existence of a so-called spanning triangle factor in G ,that is, a collection of ⌊ n/ ⌋ vertex disjoint triangles. A well-knownconjecture of P´osa (see, e.g., [6]) asserts that roughly the same min-imum degree actually guarantees the existence of a connected super-graph of a spanning triangle factor. It states that any graph G with δ ( G ) ≥ n contains a spanning squared cycle C n , where the square ofa graph, F , is obtained from F by adding edges between all pairs ofvertices with distance 2 in F . This can be seen as a 3-chromatic ana-logue of Dirac’s theorem, which turned out to be much more difficultthan its 2-chromatic cousin.Fan and Kierstead [7] proved an approximate version of P´osa’s con-jecture for large n . In addition they determined a sufficient and bestpossible minimum degree condition for the case that the squared cyclein P´osa’s conjecture is replaced by a squared path P n , i.e., the squareof a spanning path P n . Theorem 1 (Fan & Kierstead [8]) . If G is a graph on n vertices withminimum degree δ ( G ) ≥ (2 n − / , then G contains a spanning squaredpath P n . The P´osa Conjecture was verified for large values of n by Koml´os,Sark¨ozy, and Szemer´edi [10]. The proof in [10] actually asserts thefollowing stronger result, which guarantees not only spanning squaredcycles but additionally squared cycles of all lengths between 3 and n that are divisible by 3. Theorem 2 (Koml´os, S´ark¨ozy & Szemer´edi [10]) . There exists an in-teger n such that for all integers n > n any graph G of order n andminimum degree δ ( G ) ≥ n contains all squared cycles C ℓ ⊆ G with ≤ ℓ ≤ n . If furthermore K ⊆ G , then C ℓ ⊆ G for any ≤ ℓ ≤ n with ℓ = 5 . ETWEEN TUR ´AN’S THEOREM AND P ´OSA’S CONJECTURE 3
For squared cycles C ℓ with ℓ not divisible by 3 the additional con-dition K ⊆ G is necessary because these target graphs are not 3-colourable and hence a complete 3-partite graph shows that one cannothope to force C ℓ unless δ ( G ) ≥ (2 n + 1) /
3. If δ ( G ) ≥ (2 n + 1) /
3, on theother hand, then Tur´an’s Theorem asserts that G contains a copy of K and hence Theorem 2 implies C ℓ ⊆ G for any 3 ≤ ℓ ≤ n with l = 5.The case ℓ = 5 has to be excluded because C is the 5-chromatic K .In this paper we address the question of what happens between thesetwo extrema of target graphs with constant order and order n . We areinterested in essentially best possible minimum degree conditions thatenforce subgraphs covering a certain percentage of the host graph.Let us start with a simple example. It is easy to see that everygraph G with minimum degree δ ( G ) ≥ δ for 0 ≤ δ ≤ n has a matchingcovering at least 2 δ vertices (see Proposition 12( a )). This gives a lineardependence between the forced size of a matching in the host graph andits minimum degree. A more general form of the result of Corr´adi andHajnal [3] mentioned earlier is a variant of this linear dependence fortriangle factors. Theorem 3 (Corr´adi & Hajnal [3]) . Let G be a graph on n verticeswith minimum degree δ ( G ) = δ ∈ [ n, n ] . Then G contains δ − n vertex disjoint triangles. The main theorem of this paper is a corresponding result mediatingbetween Tur´an’s theorem and P´osa’s conjecture. More precisely, ouraim is to provide exact minimum degree thresholds for the appearanceof a squared path P ℓ and a squared cycle C ℓ .There are at least two reasonable guesses one might make as to whatminimum degree δ ( G ) = δ will guarantee which length ℓ = ℓ ( n, δ ) ofsquared path (or longest squared cycle). On the one hand, the degreethreshold for a spanning squared path or cycle and for a spanningtriangle factor are approximately the same. So perhaps this remainstrue for smaller ℓ : in light of Theorem 3 one could expect that ℓ ( n, δ )were roughly 3(2 δ ( G ) − n ). This turns out to be far too optimistic.On the other hand, proofs of preceding results dealing with spanningsubgraphs essentially combine greedy techniques with local changes.They simply start to construct the desired subgraph in (almost) anylocation, and in the event of getting stuck change only a few of thevertices embedded so far; at no time do they scrap an entire half-constructed object and start anew. It would not be unreasonable tobelieve that this technique also leads to best possible minimum de-gree conditions for large but not spanning subgraphs. Clearly, in thecase of (unsquared) paths such a greedy strategy provides a path oflength δ ( G ) + 1. As G might be disconnected, however, it cannot guar-antee longer paths if δ ( G ) < n/
2. For squared paths the following
PETER ALLEN, JULIA B ¨OTTCHER, AND JAN HLADK ´Y construction shows that with an arbitrary starting location one can-not hope for squared paths on more than (2 δ ( G ) − n ) vertices: If G contains disjoint cliques C and C ′ of orders 2 δ − n and n − δ , and anindependent set I of order n − δ such that all vertices of C and C ′ are connected to all vertices of I but not to other vertices of G , thenit is not difficult to see that the longest squared path in G startingin an edge of C has length (2 δ ( G ) − n ). This could lead to the ideathat ℓ ( n, δ ) were approximately (2 δ ( G ) − n ). It is true that thereare squared paths of this length in G —but this lower bound is almostalways excessively pessimistic. In other words, it turns out that onehas to carefully choose the ‘region’ of G to look for the desired squaredpath. Since spanning squared paths use all vertices of G this problemdoes not occur for these subgraphs.For fixed n both guesses propose a linear dependence between δ andthe length ℓ ( n, δ ) of a forced squared path (or cycle). As we will seebelow ℓ ( n, δ ) as a function of δ behaves very differently: it is piece-wise linear but jumps at certain points. (These jumps can be viewedas phase transitions for the appearance of squared paths or cycles.)To make this precise we introduce the following functions. Given twopositive integers n and δ with δ ∈ ( n, n − r p ( n, δ ) to bethe largest integer r such that n − δ + ⌊ δ/r ⌋ > δ and r c ( n, δ ) to be thelargest integer r such that n − δ + ⌈ δ/r ⌉ > δ . We then definesp( n, δ ) := min n l ⌈ δ/r p ( n, δ ) ⌉ + m , n o , andsc( n, δ ) := min n j ⌈ δ/r c ( n, δ ) ⌉ k , n o . (1)Observe that sc( n, δ ) ≤ sp( n, δ ) and that for almost every α ∈ (0 , n →∞ sc( n, αn ) /n = lim n →∞ sp( n, αn ) /n . The dependencebetween sp( n, δ ) and δ is illustrated in Figure 1.Our main theorem now states states that sp( n, δ ) and sc( n, δ ) arethe maximal lengths of squared paths and cycles, respectively, forcedin an n -vertex graph G with minimum degree δ . More generally, and inaccordance with Theorem 2, we show that G also contains any shortersquared cycle with length divisible by 3 (see ( i ) of Theorem 4). Weshall show below that these results are tight by explicitly constructingextremal graphs G p ( n, δ ) and G c ( n, δ ) for squared paths and cycles.While the extremal graphs of all previously discussed results are Tur´angraphs (complete r -partite graphs, where r = 3 in the case of squaredpaths and cycles) the graphs G p ( n, δ ) and G c ( n, δ ) have a rather dif-ferent structure. In fact they do contain squared cycles C ℓ for all3 ≤ ℓ ≤ sc( n, δ ) with ℓ = 5. If any one of these ‘extra’ squared cycleswith chromatic number 4 is not present in the host graph G , then ( ii )of Theorem 4 guarantees even much longer squared cycles C ℓ in G ,where ℓ is a multiple of 3. ETWEEN TUR ´AN’S THEOREM AND P ´OSA’S CONJECTURE 5
PSfrag replacements δ (2 δ − n )4 δ − n δ − n sp( n, δ ) n n n n n n n n
11 5 n n n Figure 1.
The behaviour of sp( n, δ ). Theorem 4.
For any ν > there exists an integer n such that forall integers n > n and δ ∈ [( + ν ) n, n ] the following holds for all n -vertex graphs G with minimum degree δ ( G ) ≥ δ . ( i ) P n,δ ) ⊆ G and C ℓ ⊆ G for every ℓ ∈ N with ≤ ℓ ≤ sc( n, δ ) such that divides ℓ . ( ii ) Either C ℓ ⊆ G for every ℓ ∈ N with ≤ ℓ ≤ sc( n, δ ) and ℓ = 5 ,or C ℓ ⊆ G for every ℓ ∈ N with ≤ ℓ ≤ δ − n − νn such that divides ℓ . The proof of this result relies on Szemer´edi’s Regularity Lemma andis presented together with the main lemmas in Section 2. Theorem 4cannot be extended to all values of δ ( G ) with δ ( G ) − n = o ( n ) becausefor infinitely many values of m there are C -free graphs F on m verticeswith δ ( F ) ≥ √ m (see [18]). Then, letting G be the n -vertex graphobtained from F by adding an independent set I on m −⌊ √ m ⌋ verticesand inserting all edges between F and I , it is easy to see that δ ( G ) > n + √ n but G does not contain a copy of C .The following extremal graphs show that the bounds in ( i ) and ( ii )of Theorem 4 are tight (see also Figure 2). For ( ii ) consider thecomplete tripartite graph K n − δ,n − δ, δ − n . Clearly, this graph has min-imum degree δ and does not contain C ℓ for any ℓ ≥ ℓ ≥ δ − n ). For the first part of ( i ), let G p ( n, δ ) We refer to [14] for a survey on applications of the Regularity Lemma on graphembedding problems.
PETER ALLEN, JULIA B ¨OTTCHER, AND JAN HLADK ´Y be the n -vertex graph obtained from the disjoint union of an inde-pendent set Y on n − δ vertices and r := r p ( n, δ ) cliques X , . . . , X r with | X | ≤ · · · ≤ | X r | ≤ | X | + 1 on a total of δ vertices, by insertingall edges between Y and X i for each i ∈ [ r ]. It is easy to check that δ ( G p ( n, δ )) = δ . Moreover any squared path P m ⊆ G p ( n, δ ) containsvertices from at most one clique X i . As Y is independent and P m has independence number ⌈ m/ ⌉ we have ⌊ m/ ⌋ ≤ ⌈ δ/r p ( n, δ ) ⌉ andthus m ≤ ⌊ (3 ⌈ δ/r p ( n, δ ) ⌉ + 1) ⌋ = sp( n, δ ). For the second part of ( i ), we construct the graph G ′ c ( n, δ ) in the same way as G p ( n, δ )but with r := r c ( n, δ ) and with | X i | = ⌈ δ/r ⌉ for all i ∈ [ r ]. To obtainan n -vertex graph G c ( n, δ ) from G ′ c ( n, δ ) choose v i in X i arbitrarily foreach i ∈ [ r ] and identify all v i with i ≤ r ⌈ δ/r ⌉ − δ . Again G c ( n, δ ) hasminimum degree δ , any squared cycle C m in G c ( n, δ ) touches only oneof the X i , and hence m ≤ sc( n, δ ).PSfrag replacements n − δ n − δn − δ n − δ G p ( n, δ ) G c ( n, δ ) δ − n K n − δ,n − δ, δ − n Figure 2.
The extremal graphs, for the case r p ( n, δ ) = r c ( n, δ ) = 4.Before closing this introduction let us remark that similar phenom-ena to those described in Theorem 4 are observed with simple pathsand cycles. Every graph with minimum degree δ contains a path oflength ⌈ n/ ⌊ n/ ( δ + 1) ⌋⌉ , and the extremal graph is a vertex disjointunion of cliques. This follows from an easy adjustment of the proofof Dirac’s theorem. Improving on results of Nikiforov and Schelp [17]the first author proved the following theorem in [1]. The methods usedfor obtaining this result are quite different from those applied in thispaper. In particular they do not rely on the Regularity Lemma. Theorem 5 (Allen [1]) . Given an integer k ≥ there is n such thatwhenever n ≥ n and G is an n -vertex graph with minimum degree δ ≥ n/k , the following are true. ( i ) G contains C t for every even ≤ t ≤ ⌈ n/ ( k − ⌉ , ( ii ) if G does not contain a cycle of every length from ⌊ n/δ ⌋ − to ⌈ n/ ( k − ⌉ inclusive then G does contain C t for every even ≤ t ≤ δ . ETWEEN TUR ´AN’S THEOREM AND P ´OSA’S CONJECTURE 7 Main lemmas and proof of Theorem 4
Our proof of Theorem 4 combines the Stability Method pioneeredby Simonovits [19], the Regularity Method which pivots around thejoint application of Szemer´edi’s celebrated Regularity Lemma [20], andthe so-called Blow-up Lemma by Koml´os, S´ark¨ozy and Szemer´edi [11].The combination of these three methods has proved useful for a va-riety of exact embedding results and was applied for example in [10].However, this well-established technique provides only a rather looseframework for proofs of this kind. For our application we will embel-lish this framework with a new concept, which we call the connectedtriangle components of a graph.In this section we explain how we use connected triangle components,the Regularity Method, and the Stability Method. We first provide thenecessary definitions, formulate our main lemmas (whose proofs areprovided in the remaining sections of this paper), and sketch how theywork together in the proof of Theorem 4. The details of this proof arethen presented at the end of this section.
Notation.
For a graph G we write V ( G ) and E ( G ) to denote its vertexset and edge set, respectively, and set v ( G ) = | V ( G ) | , e ( G ) = | E ( G ) | and e ( X, Y ) = |{ xy ∈ E ( G ) : x ∈ X, y ∈ Y }| for sets X, Y ⊆ V ( G ).The graph G [ X ] is the subgraph of G induced by X . The neighbour-hood of a vertex v in G is denoted by Γ( v ) and Γ( u, v ) is the commonneighbourhood of u, v ∈ V ( G ). For an edge uv = e ∈ E ( G ) we alsowrite Γ( e ) = Γ( u, v ). The minimum degree of G is denoted by δ ( G )and for two sets X, Y ⊆ V ( G ) we define δ Y ( X ) = min x ∈ X | Γ( x ) ∩ Y | and δ G ( X ) = δ V ( G ) ( X ).When we say that a statement S ( ǫ, ǫ ′ ) holds for positive real numbers ε ≫ ε ′ , then we mean that, given an arbitrary ε >
0, we can find an ǫ ′′ > S ( ǫ, ǫ ′ ) holds for all ǫ ′ ∈ (0 , ǫ ′′ ]. Connected triangle components and triangle factors.
Connectedtriangle components and connected triangle factors are the main pro-tagonists in the proof of Theorem 4. Roughly speaking, in a connectedtriangle component we can start in an arbitrary triangle and reach eachother triangle by “walking” through a sequence of triangles, and a con-nected triangle factor is a collection of vertex disjoint triangles eachpair of which is connected in this way.To make this precise, let G = ( V, E ) be a graph. A triangle walk in G is a sequence of edges e , . . . , e p in G such that e i and e i +1 sharea triangle of G for all i ∈ [ p − e and e p are triangleconnected in G . A triangle component of G is a maximal set of edges C ⊆ E such that every pair of edges in C is triangle connected. Observethat this induces an equivalence relation on the edges of G , but avertex may be part of many triangle components. In addition a trianglecomponent does not need to form an induced subgraph of G in general. PETER ALLEN, JULIA B ¨OTTCHER, AND JAN HLADK ´Y
The vertices of a triangle component C i are all vertices v such that someedge uv of G is contained in C i . We define the size | C | of a trianglecomponent C to be the number of vertices of C .A triangle factor T in a graph G is a collection of vertex disjointtriangles in G . T is a connected triangle factor if all edges of T are inthe same triangle component of G . We define the size of T to be thenumber of vertices covered by T . We let CTF( G ) denote the maximumsize of a connected triangle factor in G . It is not difficult to check forexample that any connected triangle factor in G p ( n, δ ) contains onlyvertices of at most one of the cliques X i (cf. the definition of G p ( n, δ )below Theorem 4) and of the independent set Y . Hence(2) CTF (cid:0) G p ( n, δ ) (cid:1) = 3 (cid:22) sp( n, δ )3 (cid:23) . Suppose that a graph G contains a square-path of length ℓ . Thenobviously, CTF( G ) ≥ ⌊ ℓ/ ⌋ . Thus, (2) together with Theorem 4( i )says that G p ( n, δ ) minimises CTF among all graphs of order n andminimum degree δ .We will usually find that the number of vertices in a triangle com-ponent and the size of a maximum connected triangle factor in thatcomponent are quite different. As we will explain next, for the pur-poses of embedding squared paths and squared cycles, it is the size ofa connected triangle factor that is important. The Regularity Method.
The Regularity Lemma provides a parti-tion of a dense graph that is suitable for an application of the Blow-upLemma, which is an embedding result for large host graphs. In orderto formulate the versions of these two lemmas that we will use, we firstintroduce some terminology.Let G = ( V, E ) be a graph and ε, d ∈ (0 , U, W ⊆ V the density of the pair ( U, W ) is d ( U, W ) = e ( U, W ) / | U || W | .A pair ( U, W ) is ε -regular if | d ( U ′ , W ′ ) − d ( U, W ) | ≤ ε for all U ′ ⊆ U and W ′ ⊆ W with | U ′ | ≥ ε | U | and | W ′ | ≥ ε | W | . An ε -regular partition of G is a partition V ˙ ∪ V ˙ ∪ . . . ˙ ∪ V k of V with | V | ≤ ε | V | , | V i | = | V j | forall i, j ∈ [ k ], and such that for all but at most εk pairs ( i, j ) ∈ [ k ] ,the pair ( V i , V j ) is ε -regular.Given some 0 < d < V i , V j ) in agraph G , we say that ( V i , V j ) is ( ε, d ) -regular if it is ε -regular and hasdensity at least d . We say that an ε -regular partition V ˙ ∪ V ˙ ∪ . . . ˙ ∪ V k of a graph G is an ( ε, d ) -regular partition if the following is true. Forevery 1 ≤ i ≤ k , and every vertex v ∈ V i , there are at most ( ε + d ) n edges incident to v which are not contained in ( ε, d )-regular pairs ofthe partition.Given an ( ε, d )-regular partition V ˙ ∪ V ˙ ∪ . . . ˙ ∪ V k of a graph G , wedefine a graph R , called the reduced graph of the partition of G , where ETWEEN TUR ´AN’S THEOREM AND P ´OSA’S CONJECTURE 9 R = ( V ( R ) , E ( R )) has V ( R ) = { V , . . . , V k } and V i V j ∈ E ( R ) when-ever ( V i , V j ) is an ( ε, d )-regular pair. We will usually omit the partition,and simply say that G has ( ε, d ) -reduced graph R . We call the partitionclasses V i with i ∈ [ k ] clusters of G . Observe that our definition of thereduced graph R implies that for T ⊆ V ( R ) we can for example referto the set S T , which is a subset of V ( G ).The celebrated Szemer´edi Regularity Lemma [20] states that everylarge graph has an ε -regular partition with a bounded number of parts.Here we state the so-called degree form of this lemma (see, e.g., [13,Theorem 1.10]). Lemma 6 (Regularity Lemma, degree form) . For every ε > andevery integer m , there is m such that for every d ∈ [0 , every graph G = ( V, E ) on n ≥ k vertices has an ( ε, d ) -reduced graph R on m vertices with m ≤ m ≤ m . For our purpose it is convenient to work with even a different versionof the regularity lemma, which takes into account that we are dealingwith graphs of high minimum degree. This lemma is an easy corollaryof Lemma 6. A proof can be found, e.g., in [16, Proposition 9].
Lemma 7 (Regularity Lemma, minimum degree form) . For all ε , d , γ with < ε < d < γ < and for every m , there is m such thatevery graph G on n > m vertices with δ ( G ) ≥ γn has an ( ε, d ) -reducedgraph R on m vertices with m ≤ m ≤ m and δ ( R ) ≥ ( γ − d − ε ) m . This lemma asserts that the reduced graph R of G “inherits” the highminimum degree of G . We shall use this property in order to reducethe original problem of finding a squared path (or cycle) in an n -vertexgraph with minimum degree γn to the problem of finding an arbitrary connected triangle factor of a certain size in an m -vertex graph R withminimum degree ( γ − d − ε ) m . The new problem is much less particularabout the required subgraph than the original one and hence easier toattack (see Lemma 9).This kind of reduction is made possible by the Blow-up Lemma.Roughly, this lemma asserts that a bounded degree graph H can beembedded into a graph G with reduced graph R if there is a homo-morphism from H to a subgraph S of R which does not “overfill” anyof the clusters in S . In our setting we apply this lemma with S = K and conclude that for each triangle t of a connected triangle factor T in R we find a squared path in G that almost fills the clusters of G corresponding to t . By using the fact that T is triangle connected itis then possible to connect these squared paths into squared paths orcycles of the desired overall length. In addition, the Blow-up Lemmaallows for some control about the start- and end-vertices of the paththat is constructed in this way (cf. Lemma 8( iii )). The following lemma summarises this embedding technique, whichis also implicit, e.g., in [10]. For completeness we provide a proof ofthis lemma in the appendix.
Lemma 8 (Embedding Lemma) . For all d > there exists ε el > with the following property. Given < ε < ε el , for every m el ∈ N there exists n el ∈ N such that the following hold for any graph G on n ≥ n el vertices with ( ε, d ) -reduced graph R ′ on m ≤ m el vertices. ( i ) C ℓ ⊆ G for every ℓ ∈ N with ℓ ≤ (1 − d ) CTF( R ′ ) nm . ( ii ) If K ⊆ C for each triangle component C of R ′ , then C ℓ ⊆ G for every ℓ ∈ N \ { } with ≤ ℓ ≤ (1 − d ) CTF( R ′ ) nm .Furthermore, let T be a connected triangle factor in a triangle compo-nent C of R with K ⊆ C , let u v , u v ∈ E ( G ) be disjoint edges, andsuppose that there are (not necessarily disjoint) edges X Y , X Y ∈ C such that the edge u i v i has at least d nm common neighbours in eachcluster X i and Y i for i = 1 , . Then ( iii ) P ℓ ⊆ G for every ℓ ∈ N with m + 2) < ℓ < (1 − d ) | T | nm , suchthat P ℓ starts in u , v and ends in u , v (in those orders) andat most ( ε + d ) n vertices of P ℓ are not in S T . The copies of K that are required in this lemma play a crucial rˆolewhen embedding squared cycles which are not 3-chromatic. The Stability Method.
The strategy we just described leaves us withthe task of finding a big connected triangle factor T in the reducedgraph R of G . However, there is one problem with this approach: Theproportion τ of R covered by T is roughly equal to the proportionof G covered by the squared path P that we obtain from the Embed-ding Lemma (Lemma 8). However, as explained above, the relativeminimum degree γ R = δ ( R ) / | V ( R ) | of R is in general slightly smallerthan γ G = δ ( G ) / | V ( G ) | , but the extremal graphs for squared pathsand connected triangle factors are the same. It follows that we cannotexpect that τ is larger than the proportion a maximum squared pathcovers in a graph with relative minimum degree γ R , and hence smallerthan the proportion we would like to cover for relative minimum de-gree γ G .Consequently we need to be more ambitious and shoot for a biggerconnected triangle factor in R than we can expect for this minimumdegree (cf. Lemma 9 (S1) and (S2)). This will of course not always bepossible, but it will only fail if R (and hence G ) is ‘very close’ to theextremal graph G p ( | V ( R ) | , δ ( R )) (and hence also to G c ( | V ( R ) | , δ ( R )))in which case we will say that R is near-extremal (cf. Lemma 9 (S3)).This approach is called the Stability Method and the following lemmastates that it is feasible for our purposes. This lemma additionallyguarantees copies of K as required by the Embedding Lemma. Weformulate this lemma for graphs G , but use it on the reduced graph R ETWEEN TUR ´AN’S THEOREM AND P ´OSA’S CONJECTURE 11 later. Its proof does not rely on the Regularity Lemma and is given inSection 3.
Lemma 9 (Stability Lemma) . Given µ > , for any sufficiently small η > there exists n such that if G has n > n vertices and δ ( G ) = δ ∈ (( + µ ) n, n − ) , then either (S1) CTF( G ) ≥ δ − n ) , or (S2) CTF( G ) ≥ min(sp( n, δ + ηn ) , n ) , or (S3) G has an independent set of size at least n − δ − ηn whose removaldisconnects G into components, each of size at most (2 δ − n ) .Moreover, in cases (S2) and (S3) each triangle component of G containsa K . By the discussion above, it remains to handle the graphs with near-extremal reduced graph. For these graphs we have a lot of structuralinformation which enables us to show directly that they contain thesquared paths and squared cycles we desire, as the following lemmadocuments. The proof of this lemma is provided in Section 4. In thisproof we shall again make use of the embedding lemma, Lemma 8.Accordingly Lemma 10 inherits the upper bound m el on the numberof clusters from Lemma 8. Lemma 10 (Extremal Lemma) . For every ν > , given < η, d < − ν there exists ε > such that for every < ε < ε and every m el , there exists N such that the following holds. Suppose that ( i ) G is an n -vertex graph with n > N and δ ( G ) = δ > n + νn , ( ii ) R is an ( ε, d ) -reduced graph of G of order m ≤ m el , ( iii ) each triangle component of R contains a copy of K . ( iv ) V ( R ) = I ˙ ∪ B ˙ ∪ B ˙ ∪ · · · ˙ ∪ B k with k ≥ , ( v ) I is an independent set with | I | ≥ ( n − δ − ηn ) m/n , ( vi ) for each i ∈ [ k ] we have < | B i | ≤ m (2 δ − n ) / (10 n ) , and forevery j ∈ [ k ] \ { i } there are no edges between B i and B j in R .Then G contains P n,δ ) and C ℓ for each ℓ ∈ [3 , sc( n, δ )] \ { } . It is interesting to notice that, although the two functions sp( n, δ )and sc( n, δ ) are different—their jumps as δ increases occur at slightlydifferent values—they are similar enough that the Stability Lemmacovers them both. We will only need to distinguish between squaredpaths and squared cycles when we examine the near-extremal graphs. Proof of Theorem 4.
With this we have all the ingredients for theproof of our main theorem, which uses the Regularity Lemma (in formof Lemma 7) to construct a regular partition with reduced graph R ofthe host graph G , the Stability Lemma (Lemma 9) to conclude that R either contains a big connected triangle factor or is near-extremal, theEmbedding Lemma (Lemma 8) to find long squared paths and cycles in G in the first case, and the Extremal Lemma (Lemma 10) in thesecond case. Proof of Theorem 4.
We require our constants to satisfy ν ≫ µ ≫ η ≫ d ≫ ε > , which we choose, given ν , as follows. First, we choose µ := ν/
2. Wethen choose η > d > d ≤ ν/
10 and d ≤ η/
10. For this d Lemma 8 then produces a constant ε el .We choose ε > { ε el , ν/ } and sufficientlysmall for Lemma 10. We choose m to be sufficiently large to applyLemma 9 to any graph with at least m vertices. We then choose m el such that Lemma 7 guarantees the existence of an ( ε, d )-regularpartition with at least m and at most m el parts. Finally we choose n > n el to be sufficiently large for both Lemma 8 and Lemma 10.Let n > n and δ ∈ ( n/ νn, n − G be any n -vertex graphwith δ ( G ) ≥ δ . Observe that it suffices to show that P n,δ ) ⊆ G and that ( ii ) of Theorem 4 holds. We first apply Lemma 7 to G toobtain an ( ε, d )-reduced graph R on m ≤ m ≤ m el vertices. Let δ ′ = δ ( R ) ≥ ( δ/n − d − ε ) m > m/ µm . Then we apply Lemma 9to R . There are three possibilities.First, we could find that CTF( R ) ≥ δ ′ − m ). In this case byLemma 8 we are guaranteed that for every integer ℓ ′ with 3 ℓ ′ < (1 − d ) CTF( R ) n/m we have C ℓ ′ ⊆ G . By choice of d and ε we have(1 − d ) · δ ′ − m ) n/m > δ − n − νn . Noting that P ℓ ⊆ C ℓ weconclude that P n,δ ) ⊆ G and C ℓ ⊆ G for each integer ℓ ≤ δ − n − νn such that 3 divides ℓ , i.e., the second case of Theorem 4( ii ) holds.Second, we could find that CTF( R ) ≥ min(sp( m, δ ′ + ηm ) , m ) andthat every triangle component of R contains a copy of K . By Lemma 8we are guaranteed that for every ℓ ∈ [6 , (1 − d ) CTF( R ) n/m ] \ { } wehave C ℓ ⊆ G . By choice of η and d we have (1 − d ) CTF( R ) n/m > sp( n, δ ) ≥ sc( n, δ ), so we have P n,δ ) ⊆ G and for each integer ℓ ∈ [3 , sc( n, δ )] \ { } we have C ℓ ⊆ G , i.e., the first case of Theorem 4( ii )holds.Third, we could find that R is near-extremal. Then R contains anindependent set on at least m − δ ′ − ηm vertices whose removaldisconnects R into components of size at most (2 δ ′ − m ), and eachtriangle component of R contains a copy of K . But now G satisfiesthe conditions of Lemma 10. It follows that G contains P n,δ ) and foreach ℓ ∈ [3 , sc( n, δ )] \ { } the graph G contains C ℓ , i.e., the first caseof Theorem 4( ii ) holds. (cid:3) ETWEEN TUR ´AN’S THEOREM AND P ´OSA’S CONJECTURE 13 Triangle Components and the proof of Lemma 9
In this section we provide a proof of our stability result for con-nected triangle factors, Lemma 9. Distinguishing different cases, weanalyse the sizes and the structure of the triangle components in thegraph G under study. Before we give more details about our strategyand a sketch of the proof, we introduce some additional definitions andprovide a preparatory lemma (Lemma 11).Let G be a graph with triangle components C , . . . , C r . The interior int( G ) of G is the set of vertices of G which are in more than one of thetriangle components. For a component C i , the interior of C i , writtenint( C i ), is the set of vertices of C i which are in int( G ). The remainingvertices of C i are called the exterior ∂ ( C i ). That is, ∂ ( C i ) is formedby the set of vertices of C i which are in no other triangle componentof G . To give an example, by definition the graph G p ( n, δ ) has r p ( n, δ )triangle components; its interior is the independent set Y (using thenotation of the construction of G p ( n, δ ) on page 6 in Section 1), withthe component exteriors being the cliques X , . . . , X r .The following lemma collects some observations about triangle com-ponents. Lemma 11.
Let G be an n -vertex graph with δ ( G ) = δ > n/ . Then ( a ) each triangle component C of G satisfies | C | > δ , ( b ) for distinct triangle components C , C ′ we have e ( ∂ ( C ) , ∂ ( C ′ )) = 0 , ( c ) for each triangle component C , each vertex u of C , and U := { v : uv ∈ C } , the minimum degree in G [ U ] is at least δ − n andhence | G [ U ] | ≥ δ − n + 1 .Proof. To see ( a ) let M be the vertices of a maximal clique in C (clearly | M | ≥ u and v are in M , and x is a common neighbour of u and v ,then x is also in C . Thus vertices of G \ C are adjacent to at most 1vertex of M and vertices of C are adjacent to at most | M | − M , by maximality of M . This gives the inequality | M | δ ≤ X m ∈ M d ( m ) ≤ X x ∈ C ( | M | −
1) + X x/ ∈ C | M | δ − n ≤ ( | M | − | C | . Since n < δ we have | C | > δ asrequired.Since δ > n/
2, we have that Γ( u, u ′ ) = ∅ for any two vertices u and u ′ . Now, if u ∈ ∂ ( C ), u ′ ∈ ∂ ( C ′ ), x ∈ Γ( u, u ′ ), and uu ′ was anedge, then uu ′ x would form a triangle. Then u and u ′ would be togetherin some triangle component C ′′ , contradicting the fact that they are inthe exterior. Therefore, the assertion ( b ) follows.Moreover, for an edge uv of C we have Γ( u, v ) ⊆ C as C is a trianglecomponent. Since | Γ( u, v ) | ≥ δ − n we get ( c ). (cid:3) Now let us sketch the proof of Lemma 9. Lemma 11( a ) states thattriangle components cannot be too small. However, it is not solelythe size of the triangle components we are interested in: we want tofind a triangle component that contains many vertex disjoint triangles.At this point, Lemma 11( c ) comes into play. It asserts that certainspots in a triangle component induce a graph with minimum degree2 δ − n . In the proof of Lemma 9 we shall usually (i.e., for many valuesof δ ) use this fact in order to find a big matching M in such spots(Proposition 12( a ) below asserts that this is possible). Clearly all edgesin such a matching are triangle connected and hence it will remain toextend M to a set of vertex disjoint triangles. For this purpose we willanalyse the size of the common neighbourhood Γ( u, v ) of an edge uv in M . We will usually find that Γ( u, v ) is so big that a simple greedystrategy allows us to construct the triangles. For estimating Γ( u, v )we will often use the following technique: We find a large set X suchthat neither u nor v has neighbours in X . This implies | Γ( u, v ) | ≥ δ − ( n − | X | ). Observe that Lemma 11( b ) implies that ∂ ( C ) can serveas X if both u , v ∈ ∂ ( C ′ ) for some triangle components C and C ′ .The strategy we just described works for most values of δ below n (we describe the exceptions below). For δ ≥ n however, the greedytype argument fails, the reason being that we usually bound the com-mon neighbourhood of an edge used in the argument above by 4 δ − n .But for δ ≥ n we might have sp( n, δ ) > δ − n (see Figure 1). Wesolve this problem by using a different strategy in this range of δ . Wewill still start with a big connected matching M as before, but use aHall-type argument to extend M to a triangle factor T . More precisely,we find M in the exterior of some triangle component and then con-sider for each edge uv of M all common neighbours of uv in int( G ).The Hall-type argument then permits us to find distinct extensions forthe edges of M . To make this argument work we use the fact that inthis range of δ the set int( G ) is an independent set.We indicated earlier that there are some exceptional values of δ thatrequire special treatment: namely δ close to n and n . Observe thatin both ranges the number of triangle components of G p ( n, δ ) changes(from 2 to 3 for n , and from 3 to 4 for n ) and thus the value sp( n, δ ) asa function in δ jumps (see Figure 1). Roughly speaking, the reason thatthese two ranges need to be treated separately is that again sp( n, δ ) isnot substantially smaller than 4 δ − n here, but we also do not knownow that int( G ) is an independent set. For dealing with these valuesof δ we will use a somewhat technical case analysis which we provideat the end of this section (as proof of Fact 17).As explained above, we will apply the following simple observationsabout matchings in graphs of given minimum degree. ETWEEN TUR ´AN’S THEOREM AND P ´OSA’S CONJECTURE 15
Proposition 12. ( a ) Let G = ( X, E ) be a graph with minimum degree δ . Then G hasa matching covering δ, ⌊| X | / ⌋ ) vertices. ( b ) Let G = ( A ˙ ∪ B, E ) be a bipartite graph with parts A and B , suchthat every vertex in A has degree at least a and every vertex in B has degree at least b . Then G has a matching covering a + b, | A | , | B | ) vertices.Proof. We first prove ( a ). Let M be a maximum matching in G , andassume that M contains less than min( δ, ⌊| X | / ⌋ ) edges. In particu-lar, there are two vertices x, y ∈ X not covered by M . Clearly, allneighbours of x and y are covered by M .We claim that there is an edge uv in M with xu, yv ∈ E . Indeed,suppose that this is not the case. Then | e ∩ Γ( x ) | + | e ∩ Γ( y ) | ≤ e ∈ M . We therefore have δ + δ ≤ | Γ( x ) | + | Γ( y ) | = X e ∈ M ( | e ∩ Γ( x ) | + | e ∩ Γ( y ) | ) ≤ | M | , contradicting the fact that δ > | M | .Now, let uv ∈ M be an edge as in the claim above. Since xu, yu ∈ E we get that x, u, v, y is an M -augmenting path, a contradiction.Next we prove ( b ). Let M be a maximum matching in G . We aredone unless there are vertices u ∈ A and v ∈ B not contained in M .There cannot be an edge xy ∈ M such that uy and xv are edges of G by maximality of M , since then u, y, x, v was an M -augmenting path.But u has at least a neighbours in V ( M ) ∩ B , and v at least b neighoursin V ( M ) ∩ A , so there must be at least a + b edges in M . (cid:3) Before turning to the proof of Lemma 9 let us quickly collect someanalytical data about sp( n, δ ) and r p ( n, δ ) =: r . It is not difficult tocheck that ( r + 1) n − r r + 1) − ≤ δ < rn − r + 12 r − n − δ δ − n + 1 ≤ r < δ + 12 δ − n + 1 . (3)For the proof of Lemma 9 it will be useful to note in addition thatgiven µ >
0, for every 0 < η < η = η ( µ ), there is n = n ( η ) suchthat the following holds for all n ≥ n . For all δ, δ ′ > n + µn , where δ is such that sp( n, δ + ηn ) ≤ n , and where δ ′ is such that we have r p ( n, δ ′ ) ≥ r p ( n, δ ′ ) ≥ r p ( n, δ ′ ) = r p ( n, δ ′ + ηn ), wehave(4) sp( n, δ + ηn ) ≤
32 min (cid:16) δr p ( n, δ + ηn ) − − , δ + 3 ηnr p ( n, δ + ηn ) − (cid:17) , sp( n, δ + ηn ) ≤ · δ − n ) − ≤ δ − n − ηn, andsp( n, δ ′ + ηn ) ≤ δ ′ − n, (5) which follows immediately from the definition of sp( n, δ ) in (1) (seealso Figure 1). Proof of Lemma 9.
Given µ and any 0 < η < min( , η ( µ ) , µ / η ( µ ) is as above (4), let n := max( n ( η ) , /η ) with n ( η ) asabove (4). Let n ≥ n . This in particular means that we may assumethe inequalities (4) and (5) in what follows. Define γ := δ/n , and r := r p ( n, δ ) and r ′ := r p ( n, δ + ηn ).If G has only one triangle component then Theorem 3 guaranteesthat CTF( G ) ≥ δ − n and so we are in Case (S1). Thus we mayassume in the following that G has at least two triangle components.Then Lemma 11( a ) implies that int( C ) = ∅ for any triangle compo-nent C .Suppose that C is a triangle component of G which does not containa copy of K . Let u be a vertex of C , and U := { v : uv ∈ C } . ByLemma 11( c ) we have δ ( G [ U ]) ≥ δ − n . Because C contains nocopy of K , U contains no triangle. By Tur´an’s theorem we have | U | ≥ δ − n ), and so by Proposition 12( a ) the set U contains a matching M with 2 δ − n edges. Finally we choose greedily for each e ∈ M a distinctvertex v ∈ V ( G ) such that ev is a triangle. Since U is triangle free allthese vertices must lie outside U , and since | Γ( e ) | ≥ δ − n we cannotfail to find distinct vertices for each edge. This yields a set T of 2 δ − n vertex-disjoint triangles which are all in C . So CTF( G ) ≥ δ − n and we are in case (S1). Henceforth we assume that every trianglecomponent of G contains a copy of K .We continue by considering the case n − ≤ δ < n − . The followingobservation readily implies the lemma in this range, as we will see inFact 14. Fact 13. If δ ( G ) ≥ ( − η ) n , G has exactly triangle components, int( G ) is independent, and either | int( G ) | < n − δ − ηn or the exterior X of the triangle component with most vertices satisfies | X | ≥ (2 δ − n ) , then CTF( G ) ≥ min(sp( n, δ + ηn ) , n ) .Proof of Fact 13. First, by Lemma 11( b ) a vertex x ∈ X cannot haveneighbours in the exterior of the other triangle component, so Γ( x ) ⊆ X ∪ int( G ). Thus δ ( G [ X ]) ≥ δ − | int( G ) | , which by Proposition 12( a )means that there is a matching M in G [ X ] with(6) | M | = min( δ − | int( G ) | , ⌊| X | / ⌋ )edges.We aim to pair off edges of M with vertices of int( G ) to form asufficiently large number of vertex-disjoint triangles. To see that atriangle factor resulting from this process will be connected, observethat all edges of M are in X , and since X is a triangle componentexterior, the edges of M are triangle connected. To form triangles fromedges of M and vertices of int( G ), we introduce an auxiliary bipartite ETWEEN TUR ´AN’S THEOREM AND P ´OSA’S CONJECTURE 17 graph H with vertex set M ˙ ∪ int( G ), where uv ∈ M is adjacent in H to w ∈ int( G ) iff uvw is a triangle of G . Every vertex of X has atleast δ − | X | neighbours in int( G ), and so every edge of M has at least a := 2( δ − | X | ) − | int( G ) | common neighbours in int( G ). At the sametime, since int( G ) is independent, every vertex of int( G ) has at least δ − ( n − | int( G ) | − | X | ) neighbours in X , of which all but | X | − | M | must be in M . So every vertex of int( G ) must have at least b := δ − ( n −| int( G ) |−| X | ) − ( | X |− | M | ) −| M | = δ − n + | int( G ) | + | M | edges of M in its neighbourhood. By Proposition 12( b ) there is amatching in H on at least min( a + b, | M | , | int( G ) | ) edges, and hence aconnected triangle factor in G with so many triangles. Observe that a + b = 2 δ − | X | − | int( G ) | + δ − n + | int( G ) | + | M | = 3 δ − n − | X | + | M | . (7)Since there are two triangle components in G , there is a vertex u in a triangle component exterior which is not X . Therefore u has noneighbour in X , so | X | < n − δ . Since δ ≥ ( − η ) n , by (7) we have(8) a + b > | M | − ηn . Furthermore,(9) if | X | ≤ ( − η ) n , then a + b ≥ | M | . By Lemma 11( a ) we have | int( G ) | ≥ δ − n ≥ n − ηn . Since η ≤ we have 3 | int( G ) | ≥ n − ηn > n . Thus we have CTF( G ) ≥ n if we find a matching in H coveringint( G ). It remains, then, to check that we have(10) 3 min( a + b, | M | ) ≥ min(sp( n, δ + ηn ) , n ) . We distinguish two cases.
Case 1: a + b < | M | . By (9) this forces | X | > ( − η ) n . Since we have | M | = min( δ − | int( G ) | , ⌊| X | / ⌋ ) by (6), there are two possibilities. If | M | = ⌊| X | / ⌋ then we have a + b (8) ≥ j | X | k − ηn > n − ηn > n , which proves (10) in this subcase. If, on the other hand, | M | = δ − | int( G ) | , then we use that int( G ) is independent, which impliesint( G ) ≤ n − δ and thus a + b (8) ≥ | M | − ηn = δ − | int( G ) | − ηn ≥ δ − n − ηn (5) ≥ sp( n, δ + ηn ) , which proves (10) in this subcase. Case 2: a + b ≥ | M | . In this case, H contains a matching of size | M | ,so we have CTF( G ) ≥ | M | = 3 min( δ − | int( G ) | , ⌊| X | / ⌋ ). Againthere are two possibilities, depending on | M | . If | M | = δ − | int( G ) | , weare done by (5) exactly as before. If, on the other hand, | M | = ⌊| X | / ⌋ ,then (10) holds (and hence we are done) unless(11) 3 ⌊ | X | ⌋ < min (cid:0) sp( n, δ + ηn ) , n (cid:1) . We now assume (11) in order to derive a contradiction, and make afinal subcase distinction.First assume that sp( n, δ + ηn ) < n . Then r ′ ≥ | X | < ( δ + ηn ) + 3 < δ < (2 δ − n ) , because δ ≥ ( − η ) n and η ≤ . Furthermore, since G has twotriangle components whose exterior is of size at most X by assumptionwe have | int( G ) | > n − | X | = n − δ − ηn −
6, a contradiction to thethe conditions of Fact 13.Now assume that sp( n, δ + ηn ) ≥ n . Then we have δ > ( − η ) n .By Lemma 11( a ) we have | X | ≤ n − δ < ( +2 η ) n and so | X | < (2 δ − n ). Further | int( G ) | ≥ n − | X | ≥ δ − n > n − ηn > n − δ − ηn ,which again contradicts the conditions of Fact 13. (cid:3) Fact 14.
Lemma 9 is true for n − ≤ δ < n − .Proof of Fact 14. Observe that in this range r = 2. Assume G has anedge uv in int( G ), let x be a common neighbour of u and v and C be the triangle component containing ux and vx . Since uv ∈ int( G )there are edges uy and vz of G outside C . The sets Γ( u, y ), Γ( v, z ) and { u, v, x, y, z } are pairwise disjoint, and x is not adjacent to Γ( u, y ) ∪ Γ( v, z ) ∪ { y, z } . So δ ≤ d ( x ) ≤ ( n − − δ − n ) − δ ≤ (3 n − /
5, a contradiction. Thus int( G )is an independent set, which implies | int( G ) | ≤ n − δ . Hence, byLemma 11( a ), G cannot have more than two triangle components. Inparticular, all vertices in int( G ) lie in both triangle components of G .So if | int( G ) | ≥ n − δ − ηn then int( G ) is the desired large indepen-dent set for Case (S3). If moreover all triangle component exteriorsare of size (2 δ − n ) at most we are in Case (S3). Otherwise (ifint( G ) is small or a triangle component exterior is large) Fact 13 givesCTF( G ) ≥ min(sp( n, δ + ηn ) , n ) which is Case (S2). (cid:3) For the remainder of the proof, we suppose δ < n − and accordingly r ≥ r ′ ≥
2. For dealing with this case we first establish twoauxiliary facts. The first one captures the greedy technique for findinga large connected triangle factor that we sketched in the beginning ofthis section. We will use this technique throughout the rest of theproof.
ETWEEN TUR ´AN’S THEOREM AND P ´OSA’S CONJECTURE 19
Fact 15.
If there are two sets U , U ⊆ V ( G ) such that no vertex in U has a neighbour in U , all edges in G [ U ] are triangle connected and δ ( G [ U ]) ≥ δ then CTF( G ) ≥ min(3 ⌊| U | / ⌋ , δ , δ − n + | U | ) .Proof of Fact 15. By Proposition 12( a ) we can find a matching M ′ in U covering min(2 ⌊| U | / ⌋ , δ )vertices. Let M be a subset of M ′ covering min(2 ⌊| U | / ⌋ , δ , (4 δ − n + 2 | U | ) /
3) vertices. For each edge e ∈ M in turn we pick greedilya common neighbour of e outside both M and the previously chosencommon neighbours to obtain a set T of disjoint triangles. For any x, y ∈ U we have | Γ( x, y ) | ≥ δ − ( n − | U | ). We claim that thisimplies that T can be constructed, covering all of M . Indeed, in eachstep of the greedy procedure we have strictly more than 2 δ − ( n −| U | ) − | M | ≥ e ∈ M available. Hence T covers atleast min(3 ⌊| U | / ⌋ , δ , δ − n + | U | ) vertices. Note further that T is a connected triangle factor because all edges in G [ U ] are triangleconnected. (cid:3) Below, our goal will be to show that int( G ) is an independent set.The following fact prepares us for this step. Fact 16.
Let uv be an edge in int( G ) . Unless r ′ = 2 at least one vertex, u or v , is contained in at most r ′ − triangle components.Proof of Fact 16. Let C be the triangle component containing uv ∈ int( G ) along with the (non-empty) common neighbourhood Γ( u, v )(and perhaps some other neighbours of u or v separately). Supposethat C = C , and u is a vertex of C . Then by Lemma 11( c ), there areat least 2 δ − n + 1 neighbours x of u such that the edge ux is in C . Nowsuppose that u lies in at least r ′ − C .It follows that there is a set U u ⊆ Γ( u ) of vertices x such that ux is notin C , with | U u | ≥ ( r ′ − δ − n + 1), since no edge lies in two distincttriangle components. Suppose furthermore that v too lies in at least r ′ − C . Then there exists an analo-gously defined set U v . Since all vertices of Γ( u, v ) form triangles of C with u and v , the three sets Γ( u, v ), U u and U v are pairwise disjoint,and thus | U u ∪ U v | ≥ (2 r ′ − δ − n + 1). Now given any x ∈ Γ( u, v ),since ux and vx are both in C , x cannot be adjacent to any vertexof U u ∪ U v . But then δ ≤ d ( x ) < n − (2 r ′ − δ − n + 1) which isequivalent to 2 r ′ − < ( n − δ ) / (2 δ − n + 1). By (3) the right-hand sideis at most r and thus we get 2 r ′ − < r . Since r ≤ r ′ + 1 however thisis a contradiction unless r ′ ≤ (cid:3) We assume from now on, that(12) CTF( G ) < sp( n, δ + ηn ) , that is, we are not in Cases (S1) or (S2). Our aim is to concludethat then ( ∗ ) int( G ) is an independent set and that its vertices arecontained in at least r ′ triangle components. It turns out, however,that we need to consider the cases r = r ′ + 1 = 3 and r = r ′ + 1 = 4(i.e., the cases when the minimum degree δ is just a little bit below n and n , respectively) separately. Unfortunately these two cases, whichare treated by Fact 17, require a somewhat technical case analysis,which we prefer to defer to the end of the section. Fact 17. If r = r ′ + 1 = 3 or r = r ′ + 1 = 4 then int( G ) is anindependent set all of whose vertices are contained in at least r ′ trianglecomponents. Assuming this fact is true we can deduce ( ∗ ) for all values r ≥ Fact 18.
The set int( G ) is an independent set (and hence of size atmost n − δ ) all of whose vertices are contained in at least r ′ trianglecomponents.Proof of Fact 18. Recall that we have r ≥ r = r ′ + 1 = 3 and r = r ′ + 1 = 4 are handled byFact 17. So we assume we are not in these cases; in particular, r ′ ≥ G ) is contained in at least r ′ triangle components. Once we establish this, Fact 16 implies that thereare no edges in int( G ) and so int( G ) is an independent set as desired.To prove that each vertex of int( G ) is contained in at least r ′ trianglecomponents we assume the contrary and show that then CTF( G ) ≥ sp( n, δ + ηn ), a contradiction to (12). Indeed, let w ∈ int( G ) and sup-pose that there are k > C , . . . , C k containing w .For i ∈ [ k ] let U i be the set of neighbours u of w such that uw ∈ C i .By Lemma 11( c ) we have δ ( G [ U i ]) ≥ δ − n and | U i | ≥ δ − n + 1.Suppose that U is the largest of the U i . No vertex in U has aneighbour in U , since the components are distinct. In addition, alledges in G [ U ] are triangle connected, because U ⊆ Γ( w ). ThereforeFact 15 implies that there is a connected triangle factor T in G coveringmin(3 ⌊| U | / ⌋ , δ − n ) , δ − n + | U | ) ≥ min(3 ⌊| U | / ⌋ , δ − n ) ver-tices. If w lies only in r ′ − | U | ≥ δ/ ( r ′ − T covers at least min(3 ⌊ δ/ (2 r ′ − ⌋ , δ − n ) vertices.Now since (4) holds, we have δ/ ( r ′ − − ≥ sp( n, δ + ηn ). Since r ≥ r ′ ≥ r = r ′ + 1 = 4, by (5) we have4 δ − n ≥ sp( n, δ + ηn ). It follows that T covers at least sp( n, δ + ηn )vertices, in contradiction to (12). (cid:3) Fact 19.
We are in Case (S3) .Proof of Fact 19.
Fact 18 tells us that int( G ) is an independent set. ByLemma 11( a ) and the fact that δ > n − δ we have that every triangle ETWEEN TUR ´AN’S THEOREM AND P ´OSA’S CONJECTURE 21 component in G has an exterior, and by Lemma 11( b ) that there areno edges between any triangle component exteriors. Hence, to showthat we are in Case (S3), it is enough to prove that(13) | int( G ) | := α ≥ n − δ − ηn and | X | ≤ δ − n )for the biggest triangle component exterior X in G . Suppose for acontradiction that this is not the case. We first claim that this forces G to have exactly r ′ triangle components.Indeed, assume G has k ≥ r ′ + 1 triangle components. Each of thesecomponents C has vertices in its exterior ∂ ( C ), and so by Lemma 11( b )the minimum degree of G implies | ∂ ( C ) | ≥ δ − α +1 ≥ δ − n +1. We letthese triangle component exteriors be X , . . . , X k , with X being thebiggest. Since n = | X ˙ ∪ . . . ˙ ∪ X k ˙ ∪ int( G ) | , we have ( r ′ + 1)( δ − α ) + α
3, and since 2 δ − n ≥ µn , we have(2 δ − n + 2 ηn + 1)(1 − µ ) < δ − n + 3 ηn − µ (2 δ − n ) ≤ δ − n + 3 ηn − µ n < δ − n , and we obtain µn +(1 − µ )( n − δ − ηn ) < n − δ which is a contradictionsince n − δ < n/ η < µ .Hence, if (13) fails, then G has indeed exactly r ′ triangle components.Now we use this fact in order to derive a contradiction to (12). Ob-serve that, if r ′ = 2, and accordingly δ ≥ ( − η ) n , then Fact 13 implies that (13) holds, because according to (12) we have CTF( G ) < sp( n, δ + ηn ). In the remainder we assume r ′ ≥ X has neighbours only in X and int( G ), and | int( G ) | ≤ n − δ , we have δ ( G [ X ]) ≥ δ − n . Furthermore, sinceno vertex in X has neighbours in either X or X , and | X ˙ ∪ X | ≥ δ − n + 1), we can apply Fact 15 to obtainCTF( G ) ≥ min (cid:0) ⌊| X | / ⌋ , δ − n ) , δ − n + 2(2 δ − n + 1) (cid:1) = min (cid:0) ⌊| X | / ⌋ , δ − n ) (cid:1) . Now by (5), CTF( G ) ≥ δ − n ) is a contradiction to (12), so tocomplete our proof it remains to show that if (13) fails, then CTF( G ) ≥ ⌊| X | / ⌋ is also a contradiction to (12). Again, we distinguish twocases. Case 1: (13) fails because α < n − δ − ηn . Since X is the largestexterior, we have | X | ≥ ( δ + 11 ηn ) /r ′ . But we have by (4) thatsp( n, δ + ηn ) ≤ δ + 3 ηnr ′ − < j δ + 11 ηn r ′ k , so that CTF( G ) ≥ ⌊| X | / ⌋ is indeed a contradiction to (12). Case 2: (13) fails because | X | > (2 δ − n ). Then CTF( G ) ≥ ⌊| X | / ⌋ ≥ (2 δ − n ) −
2, which by (5) is a contradiction to (12), asdesired. (cid:3)
This completes, modulo the proof of Fact 17, the proof of Lemma 9. (cid:3)
It remains to show Fact 17. Note that we can use all facts from theproof of Lemma 9 that precede Fact 17. We will further assume thatall constants and variables are set up as in this proof.
Proof of Fact 17.
Recall that we assumed (12), i.e., CTF( G ) < sp( n, δ + ηn ), in this part of the proof of Lemma 9. We distinguish two cases. Case 1: r = 3 and r ′ = 2. In this case δ ( G ) ∈ [( − η ) n, ( + η ) n ].Trivially each vertex of int( G ) is contained in at least r ′ = 2 trianglecomponents. Suppose for a contradiction that there is an edge uv inint( G ). Let x be a common neighbour of u and v , and C be the trianglecomponent containing the triangle uvx . Let U := { y : uy ∈ C } and V := { y : vy ∈ C } and let U := Γ( u ) \ U and V := Γ( v ) \ V . Observethat U ∩ V = ∅ .By definition x is not in, and has no neighbour in, U ˙ ∪ V . It followsthat | U ˙ ∪ V | < n − δ ≤ ( +2 η ) n . On the other hand, by Lemma 11( c ),we have | U | , | V | > δ − n ≥ n − ηn , and thus | U | , | V | ∈ (cid:2) ( − η ) n, ( + 6 η ) n (cid:3) . Since d ( u ) ≥ δ ≥ ( − η ) n , we have | U | ≥ δ − | U | ≥ ( − η ) n . Butno vertex in U is adjacent to any vertex in U . This implies that everyvertex in U is adjacent to all but at most n − δ − | U | ≤ ηn vertices ETWEEN TUR ´AN’S THEOREM AND P ´OSA’S CONJECTURE 23 outside U . Since η < we have | U | > ηn , so δ ( G [ U ]) > | U | / a ), U contains a matching M u with ⌊| U | / ⌋ edges. Since each vertex of U has at most 10 ηn non-neighbours out-side U , each pair of vertices has common neighbourhood covering allbut at most 20 ηn vertices of V ( G ) \ U . In particular, the commonneighbourhood of each edge of M u covers all but at most 20 ηn ver-tices of V ( G ) \ U . Similarly, V contains a matching M v with ⌊| V | / ⌋ edges, and the common neighbourhood of each edge covers all but atmost 20 ηn vertices of V ( G ) \ V .Since 20 ηn < | U | / U ∩ V = ∅ , the common neighbourhoodof each edge of M v contains more than half of the edges of M u . Bysymmetry, the reverse is also true. Thus all edges in M u ˙ ∪ M v are inthe same triangle component of G . Finally, each edge of M u ˙ ∪ M v hasat least δ − ηn − | U ˙ ∪ V | ≥ ( − η ) n common neighbours out-side U ˙ ∪ V . Choosing greedily for each edge of M u ˙ ∪ M v in successiondistinct common neighbours outside U ˙ ∪ V , we obtain a connected tri-angle factor with min( ⌊| U | / ⌋ + ⌊| V | / ⌋ , ( − η ) n ) = ( − η ) n triangles. But then CTF( G ) ≥ ( − η ) n > n/ > sp( n, δ + ηn ),a contradiction to (12). This proves Fact 17 for the case r = 3 and r ′ = 2. Case 2: r = 4 and r ′ = 3. This implies that ( − η ) n ≤ δ ( G ) ≤ ( + η ) n , and consequently sp( n, δ + ηn ) < ( + 2 η ) n . We first provetwo statements about the structure of G which are forced by (12).(Ψ) If a vertex u has sets of neighbours U , U ′ on edges in exactly twodifferent triangle components with | U | ≥ | U ′ | then ( − η ) n < | U ′ | < ( + 6 η ) n and ( − η ) n < | U | < ( + 2 η ) n . Proof of (Ψ) . For the lower bound on | U ′ | , observe that by ( c ) ofLemma 11 we have δ ( G [ U ′ ]) ≥ δ − n ≥ ( − η ) n . To obtain theupper bound, again by Lemma 11( c ) we have δ ( G [ U ]) ≥ δ − n , andsince the sets U and U ′ are neighbours of u in different triangle compo-nents C and C ′ , there are no edges from U to U ′ . Furthermore, sinceany edge in G [ U ] forms a triangle with u using an edge from u to U ,all edges in G [ U ] are in C . Now by Fact 15 we haveCTF( G ) ≥ min(3 ⌊| U | / ⌋ , δ − n ) , δ − n + | U ′ | ) . Since | U | ≥ δ/ ⌊| U | / ⌋ ≥ ( − η ) n − > sp( n, δ + ηn ).By (5) we have 3(2 δ − n ) > sp( n, δ + ηn ). Because (12) holds, wehave 2 δ − n + | U ′ | < sp( n, δ + ηn ) < ( + 2 η ) n , and therefore | U ′ | < ( + 6 η ) n . Now the claimed lower and upper bounds on | U | follow from U = Γ( u ) \ U ′ , and from the fact that no vertex in U ′ has a neighbourin U , respectively. (cid:3) (Ξ) If a vertex u has sets of neighbours U , U , U on edges in exactlythree different triangle components then ( + 2 η ) n > | U i | > ( − η ) n for i ∈ [3]. Proof of (Ξ) . Assume that U is the largest of the three sets. By ( c )of Lemma 11 we have δ ( G [ U i ]) ≥ δ − n ≥ ( − η ) n for each i , so | U i | > ( − η ) n for each i . As in the previous case, there can be noedge from U to U ˙ ∪ U , and all edges in U are triangle-connected.Thus by Fact 15 we haveCTF( G ) ≥ min (cid:0) ⌊| U | / ⌋ , δ − n ) , δ − n + | U ˙ ∪ U | (cid:1) . Now since sp( n, δ + ηn ) < ( − η ) n and (12) holds, we have3 ⌊| U | / ⌋ < sp( n, δ + ηn ) ≤ ( 27 + 2 η ) n which implies | U | < ( + 2 η ) n . Since | U | , | U | ≤ | U | this completesthe desired upper bounds. The lower bounds follow from | U | + | U | + | U | ≥ δ ≥ ( − η ) n . (cid:3) Next we show that(Θ) int( G ) is an independent set. Proof of (Θ) . Assume for a contradiction that there is an edge uv ∈ int( G ). By Fact 16 one of the vertices of this edge, say u , is in only 2triangle components. Let its neighbours be U and U in these twotriangle components, and let the neighbours of v be partitioned intosets V , . . . , V k according to the triangle component containing the edgeto v . Assume further that Γ( u, v ) ⊆ U ∩ V , so that U , V , . . . , V k arepairwise disjoint. Let x ∈ Γ( u, v ). Since x has neighbours in neither U nor V , and since by Lemma 11( c ) we have | V | > ( − η ) n , we concludethat δ ≤ d ( x ) ≤ n − − | U | − | V | . In particular, | U | < ( − η ) n because δ ≥ ( − η ) n , and therefore by (Ψ) we have( 17 − η ) n < | U | < ( 17 + 6 η ) n . Next we want to derive analogous bounds for | V | . For this purpose wefirst show that k = 2.Indeed, if we had k = 3, then by (Ξ) d ( x ) ≤ n − − | U | − | V | − | V |≤ n − − ( − η ) n − − η ) n < ( + 16 η ) n < δ , and this contradicts δ ( G ) ≥ δ . Similarly, if k ≥
4, then by Lemma 11( c )we have | V i | ≥ ( − η ) n for each i , and hence d ( x ) ≤ n − − | U | − | V | − | V | − | V | < ( + 16 η ) n < δ , which too is a contradiction. It follows that k = 2 as claimed.Hence, we can argue analogously as before (for U ) that | V | > ( − η ) would contradict d ( x ) ≥ δ . Consequently, by (Ψ) we have( 17 − η ) n < | V | < ( 17 + 6 η ) n . ETWEEN TUR ´AN’S THEOREM AND P ´OSA’S CONJECTURE 25
We now argue that this yields a contradiction to (12) in much thesame way as we argued in the r = r ′ + 1 = 3 case. Every vertexof U is adjacent to all but at most n − | U | − δ ≤ ηn vertices of V ( G ) \ U . Since | U | > ηn , by Proposition 12( a ) there is a matching M u in U covering all but at most one vertex of U . Each edge of M u has at least δ − ηn ≥ ( − η ) n common neighbours outside U . Similarly, in V there is a matching M v covering all but at mostone vertex of V , each edge of which has at least ( − η ) n commonneighbours outside V . Since Γ( u, v ) = U ∩ V , we have U ∩ V = ∅ .It follows that every edge of M v has more than half of the edges of M u in its common neighbourhood, and thus the edges M u ˙ ∪ M v are triangleconnected. Choosing greedily for each edge in M u ˙ ∪ M v in successiona distinct common neighbour outside M u ˙ ∪ M v , we obtain a connectedtriangle factor with as many triangles as there are edges in M u ˙ ∪ M v .Since | U | , | V | > ( − η ) n , we have CTF( G ) > ( − η ) n − > sp( n, δ + ηn ), contradicting (12). This completes the proof that int( G )is an independent set. (cid:3) It remains to show that each vertex u ∈ int( G ) is contained in atleast r ′ = 3 triangle components. Assume for a contradiction thatthis is not the case and that some vertex u is only contained in 2triangle components, C and C ′ . Let U and U ′ , respectively, be theneighbours of u on edges in C and C ′ . Without loss of generality | U | ≥| U ′ | . Because int( G ) is an independent set, U and U ′ are contained inthe exteriors of C and C ′ . By Lemma 11( b ) there are thus no edgesbetween U and ∂ ( C ′ ). By Lemma 11( c ) we have δ ( G [ U ]) ≥ δ − n ,and since U ⊆ ∂ ( C ) every edge of G [ U ] is in C . It follows that we mayapply Fact 15 to obtainCTF( G ) ≥ min (cid:0) ⌊| U | / ⌋ , δ − n ) , δ − n + | ∂ ( C ′ ) | (cid:1) . Since | U | ≥ δ/ ⌊| U | / ⌋ ≥ ( − η ) n − > sp( n, δ + ηn ).By (5) we have 3(2 δ − n ) > sp( n, δ + ηn ). Since (12) holds, we concludethat 2 δ − n + | ∂ ( C ′ ) | < sp( n, δ + ηn ) < ( + 2 η ) n , and therefore | ∂ ( C ′ ) | < ( + 6 η ) n .Now any vertex in ∂ ( C ′ ) has neighbours only in ∂ ( C ′ ) ˙ ∪ int( G ), andtherefore | int( G ) | ≥ δ − | ∂ ( C ′ ) | ≥ ( − η ) n . The vertex u has neigh-bours only in U ′ ⊆ ∂ ( C ′ ) and U , and therefore | U | ≥ δ − | U ′ | ≥ δ − | ∂ ( C ′ ) | ≥ ( − η ) n . By Lemma 11( c ) we have δ ( G [ U ]) ≥ δ − n ≥ ( − η ) n , and since | U | > ( − η ) n we obtain by Proposition 12( a ) a matching M in U withat least ( − η ) n edges. Now each vertex in int( G ) is adjacent to all butat most n − δ − | int( G ) | ≤ ηn vertices outside int( G ). In particular,each vertex in int( G ) is adjacent to all but at most 10 ηn vertices of M ,and is therefore a common neighbour of all but at most 10 ηn edges of M . We now match greedily vertices of int( G ) with distinct edgesof M forming triangles. Since | int( G ) | > | M | , we will be forced tohalt only when we come to a vertex x ∈ int( G ) which is not a commonneighbour of any remaining edge of M , i.e., when we have used all butat most 10 ηn edges of M . It follows that we obtain a triangle factor T with at least ( − η ) n triangles. Since each triangle uses an edge of M ⊆ G [ U ] ⊆ G [ ∂ ( C )], T is a connected triangle factor, and we haveCTF( G ) ≥ ( − η ) n > sp( n, δ + ηn ) in contradiction to (12). (cid:3) Near-extremal graphs
In this section we provide the proof of Lemma 10. To prepare thisproof we start with two useful lemmas. The first will be used to con-struct squared paths and squared cycles from simple paths and cycles.
Lemma 20.
Given a graph G , let T = ( t , t , . . . , t l ) be a path in G and W a set of vertices disjoint from T . Let Q = ( t , t ) , Q i =( t i − , t i − , t i − , t i ) for all < i ≤ l , and Q l +1 = ( t l − , t l ) . If thereexists an ordering σ of [ l + 1] such that for each i the vertices in Q σ ( i ) have at least i common neighbours in W , then there is a squared path ( q , t , t , q , t , t , q , . . . , t ℓ , q ℓ +1 ) in G , with q i ∈ W for each i , using every vertex of T .If T is a cycle on l vertices we let instead Q = ( t l − , t l , t , t ) , Q i = ( t i − , t i − , t i − , t i ) for all < i ≤ l , and σ be an orderingon [ l ] . Then, under the same conditions, we obtain a squared cycle C l .Proof. We need only ensure that for each i one can choose q i such that q i is a common neighbour of Q i and the q i are distinct. This is possible bychoosing for each i in succession q σ ( i ) to be any so far unused commonneighbour of Q σ ( i ) . (cid:3) The second lemma is a variant on Dirac’s theorem and permits usto construct paths and cycles of desired lengths which keep some ‘bad’vertices far apart.
Lemma 21.
Let H be a graph on h vertices and B ⊆ V ( H ) be ofsize at most h/ . Suppose that every vertex in B has at least | B | neighbours in H , and every vertex outside B has at least h/ | B | +10 neighbours in H . Then for any given ≤ ℓ ≤ h we can find a cycle T ℓ of length ℓ in H on which no four consecutive vertices contain morethan one vertex of B . Furthermore, if x and y are any two vertices notin B and ≤ ℓ ≤ h , we can find an ℓ -vertex path T ℓ whose endverticesare x and y on which no four consecutive vertices contain more thanone vertex of B ∪ { x, y } .Proof. If we seek a path in H from x to y then we create a ‘dummyedge’ between x and y . If we seek a cycle, let xy be any edge of H − B . ETWEEN TUR ´AN’S THEOREM AND P ´OSA’S CONJECTURE 27
First we construct a path P in H covering B with the desired prop-erty. Let B = { b , b , . . . , b | B | } . For each 1 ≤ i ≤ | B | −
1, choose avertex u i ∈ H − B adjacent to b i and a vertex v i ∈ H − B adjacentto b i +1 . Because both u i and v i have h/ | B | + 10 neighbours in H ,they have at least 18 | B | + 20 common neighbours. At most 3 | B | ofthese are either in B or amongst the chosen u j , v j , and so we can finda so far unused vertex w i adjacent to u i and v i . Since we require only | B | − w , . . . , w | B |− we can pick the vertices greedily.We let v be yet another vertex adjacent to b , and u | B | adjacentto b | B | , and choose any further vertices w , v , w | B | , u | B | such that P = ( x, y, u , w , v , b , u , w , v , b , . . . , v | B |− , b | B | , u | B | , w | B | , v | B | )is a path on 4 | B | + 5 vertices.Now we let P ′ be a path extending P in H of maximum length. Weclaim that P ′ is in fact spanning. Suppose not: let u be an end-vertexof P ′ and v a vertex not on P ′ . Since P ′ is maximal every neighbourof u is on P ′ , so v ( P ′ ) > h/ | B | + 10. If there existed an edge u ′ v ′ of P ′ − P with u ′ u and v ′ v edges of H , with v ′ closer to u on P ′ than u ′ ,then we would have a longer path extending P in H . Counting theedges leaving u and v yields a contradiction.Finally we let u and v be the end-vertices of the spanning path P ′ .If uv is an edge of H , or if u ′ v ′ is an edge of P ′ − P , with u ′ nearerto u on P ′ than v ′ , such that uv ′ and u ′ v are edges of H , then weobtain a cycle T spanning H and containing P as a subpath. Againedge counting reveals that such an edge must exist.To obtain a cycle T ℓ with h − | B | − ≤ ℓ < h we take u to be anend-vertex of the path T − P and v its successor on T − P . If we canfind two further vertices u ′ and v ′ on T − P (in that order from u along T − P ) with h − ℓ vertices between them and with uu ′ and vv ′ edgesof H then we would obtain a cycle T ℓ of length ℓ . Again simple edgecounting reveals that such a pair of vertices exists. To obtain a cycle T ℓ with 3 ≤ ℓ < h − | B | − H − B has minimum degree h/ | B | + 10 > ( h − | B | ) / xy .The cycle T ℓ satisfies the condition that no four consecutive verticescontain more than one vertex of B , since either it preserves P as asubpath or it contains no vertices of B at all. Similarly the path from x to y within T ℓ satisfies the required conditions. (cid:3) Before embarking upon the proof of Lemma 10 we give an outlineof the method. We recall that the Szemer´edi partition supplied to theLemma is essentially the extremal structure. We shall show that theunderlying graph either also has an extremal structure, or possessesfeatures which actually lead to longer squared paths and cycles thanrequired for the conclusion of the Lemma. This is complicated by thefact that the Szemer´edi partition is insensitive both to mis-assignment of a sublinear number of vertices and to editing of a subquadratic num-ber of edges: we must assume, for example, that although the vertexset I in the reduced graph R is independent, the vertex set S I mayfail to contain some vertices of G with no neighbours in S I , and maycontain a small number of edges meeting every vertex. However, ob-serve that by the definition of an ( ε, d )-regular partition, there are novertices of S I with more than ( ε + d ) n neighbours in S I . Fortunately,it is possible to apply the following strategy in this case.We start by separating those vertices with ‘few’ neighbours in S I ,which we shall collect in a set W , and those with ‘many’. We are thenable to show (as Fact 23 below) that, if there are two vertex disjointedges in W , then the sets S B and S B are in the same triangle com-ponent of G (‘unexpectedly’, since B and B are in different trianglecomponents in R ). We shall show that in this case it is possible to con-struct very long squared paths and cycles by making use of Lemma 8.Hence we can assume that there are not two disjoint edges in W ,which in turn implies that W is almost independent and will give usrather precise control about the size of W . In addition, the minimumdegree condition will guarantee that almost every edge from W to theremainder of G is present. We would like to then say that in V ( G ) \ W we can find a long path, which together with vertices from W formsa squared path (and similarly for squared cycles). Unfortunately since G [ W, V ( G ) \ W ] is not necessarily a complete bipartite graph, thisstatement is not obviously true: although by definition no vertex out-side W has very few neighbours in W , it is certainly possible that twovertices outside W could fail to have a common neighbour in W . Butthe statement is true for a path possessing sufficiently nice properties—specifically, satisfying the conditions of Lemma 20—and the purposeof Lemma 21 is to provide paths and cycles with those nice properties.The remainder of our proof, then, consists of setting up conditions forthe application of Lemma 21. Proof of Lemma 10.
Given ν >
0, suppose the parameters η > d > η ≤ ν and d ≤ ν Given d >
0, Lemma 8 returns a constant ε el >
0. We set(15) ε = min (cid:0) ν , ε el (cid:1) . Given m el and 0 < ε < ε , Lemma 8 returns a constant n el . We set(16) N = max (cid:0) m el , η − ν − , n el (cid:1) . Now let G , R , and the partition V ( R ) = I ˙ ∪ B ˙ ∪ . . . ˙ ∪ B k satisfy condi-tions ( i )–( vi ) of the lemma. ETWEEN TUR ´AN’S THEOREM AND P ´OSA’S CONJECTURE 29 If δ ( G ) = δ ≥ n − then we can appeal to Theorem 1 to find aspanning squared path in G ; if δ ≥ n then we can appeal to Theorem 2to find C ℓ for each ℓ ∈ [3 , n ] \ { } . Therefore, the definition of sp( n, δ )and sc( n, δ ) imply that we may assume δ < n/ n/ I and of the B i . Define δ ′ = ( δ/n − d − ε ) m . Since R is an ( ε, d )-reduced graph we have(18) δ ( R ) ≥ δ ′ = ( δ/n − d − ε ) m . Observe that moreover(19) | I | ≤ m − δ ′ ≤ (cid:16) − δn + d + ε (cid:17) m , by ( v ) because clusters in I have δ ′ neighbours outside I in R . For i ∈ [ k ], fix a cluster C ∈ B i . By assumption ( vi ) C has neighboursonly in B i ∪ I in R . Since δ ′ ≤ deg( C ) = deg( C, B i ∪ I ) ≤ deg( C, B i ) + | I | ≤ deg( C, B i ) + m − δ ′ , we have | B i | > deg( C, B i ) ≥ δ ′ − m ≥ mn (cid:0) δ − dn − εn ) − n (cid:1) = mn (cid:0) δ − n − ( d + ε ) n (cid:1) . Now since 2 δ − n ≥ νn by ( i ), we conclude from (14) and (15) that(20) | B i | ≥ m (2 δ − n )3 n ≥ νm . We next show that each B i is part of exactly one triangle componentof R . Fact 22.
For each ≤ i ≤ k the following holds. All edges in R [ B i ] are triangle connected in R .Proof of Fact 22. By assumption ( vi ) we have(21) | B i | ≤ m (2 δ − n ) / (10 n ) ≤ δ ′ − m ) / , where the second inequality comes from (14) and (15). Since we have δ R ( B i ) ≥ δ ′ − m > | B i | /
2, the graph R [ B i ] is connected. It followsthat if there are two edges in R [ B i ] which are not triangle-connected,then there are two adjacent edges in R [ B i ] with this property. Thatis, there are vertices P , Q and Q ′ of B i such that P Q is in trianglecomponent C and P Q ′ is in triangle component C ′ with C = C ′ .We now show that there are at least 2 δ ′ − m edges leaving P in R [ B i ]which are in C . There are two possibilities. First, suppose there are no C -edges from P to I . In this case, the common neighbourhood Γ( P Q )must lie entirely in B i . Every vertex of Γ( P Q ) makes a C -edge with P ,and we have | Γ( P Q ) | ≥ δ ′ − m as required. Second, suppose that there is a C -edge P P ′ with P ′ ∈ I . Since I is an independent set in R ,the set Γ( P P ′ ) lies entirely within B i , and has size at least 2 δ ′ − m .Again, every edge from P to Γ( P P ′ ) is a C -edge, as desired.By the identical argument, there are at least 2 δ ′ − m edges leaving P in R [ B i ] which are in C ′ . Since no edge is in both C and C ′ , there areat least 2(2 δ ′ − m ) edges leaving P in R [ B i ], so | B i | ≥ δ ′ − m ). Thiscontradicts (21). It follows that all edges of B i are triangle connected,as desired. (cid:3) We next define a set W of those vertices in G which have few neigh-bours in S I . The intuition is that W consists of S I and only a fewmore vertices of G . To simplify notation, we set ξ = √ ε + d + 11 η .By (14) and (15), we have(22) ξ ≤ ν/ . Let W be the vertices of G which do not have more than ξn neighboursin S I . Since ξ > d + ε , by the independence of I and by the definitionof an ( ε, d )-regular partition, we have S I ⊆ W . By assumption ( v )we have | I | ≥ ( n − δ − ηn ) m/n . Hence every edge in W has at least(23) 2( δ − ξn ) − (cid:16) n − | [ I | (cid:17) > δ − (2 δ − n )16common neighbours outside S I , where we use assumption ( i ) that2 δ − n > νn , (14) and (22).By this observation, the next fact implies that we are done if thereare two vertex disjoint edges in W . Fact 23. If u v and u v are vertex disjoint edges of G such that for i = 1 , the edge u i v i has at least δ − (2 δ − n ) / common neighboursoutside S I , then G contains P n,δ ) and C ℓ for each ℓ ∈ [3 , sc( n, δ )] \{ } .Proof of Fact 23. Let D ′ be the set of clusters C ∈ V ( R ) \ I suchthat u v has at most 2 dn/m common neighbours in C . By the hy-pothesis, u v has at least δ − (2 δ − n ) /
16 common neighbours out-side S I . Of these, at most εn are in the exceptional set V of theregular partition, and at most 2 dn | D ′ | /m are in S D ′ . The remainingcommon neighbours must all lie in S ( V ( R ) \ ( I ∪ D ′ ), and hence wehave the inequality δ − δ − n − εn − dn | D ′ | m ≤ ( m − | I | − | D ′ | ) nm ( v ) ≤ n − ( n − δ − ηn ) − | D ′ | nm . Simplifying this, we obtain n − dnm | D ′ | ≤ ηn + εn + 2 δ − n , ETWEEN TUR ´AN’S THEOREM AND P ´OSA’S CONJECTURE 31 and by (14) and (15), we get | D ′ | ≤ (2 δ − n ) m/ (14 n ).Now let D be the set of clusters C ∈ V ( R ) \ I such that either u v or u v has at most 2 dn/m common neighbours in C . The sameanalysis holds for u v , so we obtain(24) | D | ≤ (2 δ − n ) m n . Therefore, we conclude from (20) that B \ D = ∅ . Take X ∈ B \ D arbitrarily. We havedeg( X, B ) ( vi ) ≥ deg( X ) − | I | ≥ δ ′ − | I | (19) ≥ δ ′ − (cid:16) − δn + d + ε (cid:17) m (18) ≥ (cid:16) δn − d − ε (cid:17) m − (cid:16) − δn + d + ε (cid:17) m (14),(15) ≥
12 (2 δ − n ) mn (24) > | D | . Thus there exists a cluster Y ∈ Γ( X ) ∩ ( B \ D ). Hence we have clusters X, Y ∈ B \ D such that XY ∈ E ( R ). Analogously, we can find clusters X ′ , Y ′ ∈ B \ D such that X ′ Y ′ ∈ E ( R ).Since δ R ( B ) , δ R ( B ) ≥ δ ′ − | I | ≥ δ ′ − m , we can find greedily amatching M in R [ B ∪ B ] with δ ′ − | I | edges. Since every clusterin I has at most m − | I | − δ ′ non-neighbours outside I , every clusterin I forms a triangle with at least | M | − ( m − | I | − δ ′ ) = 2 δ ′ − m edges of M . In addition, by assumption ( v ), (14), and since δ < n/ | I | > ( − η ) m ≥ m . Therefore we may choose greedilyclusters in I to obtain a set T of at leastmin (cid:8) δ ′ − m, | I | (cid:9) ≥ min n δ ′ − m, m o vertex-disjoint triangles formed from edges of M and clusters of I .Let T be the triangles of T contained in B ∪ I , and T those containedin B ∪ I .By Fact 22, since each triangle in T contains an edge of B , alltriangles in T are in the same triangle component as the edge XY .Similarly all the triangles in T are in the same triangle component asthe edge X ′ Y ′ .Noting that ε satisfies (15) and n > N satisfies (16), we can applyLemma 8 with X = X = X , Y = Y = Y to find a squared pathstarting with u v and finishing with u v using the triangles T . Sim-ilarly, using Lemma 8 with X = X = X ′ , Y = Y = Y ′ we find asquared path (intersecting the first only at u , v , u , and v ) start-ing with u v and finishing with u v using the triangles T . Choos-ing appropriate lengths for these squared paths and concatenatingthem we get a squared cycle C ℓ in G , for any 36( m el + 2) ≤ ℓ ≤ − d ) min { δ ′ − m, m/ } n/m . Applying Lemma 8 to the copy of K in B directly we obtain C ℓ for each ℓ ∈ [3 , n/m ] \ { } . By (16) we have 3 n/m > m el + 2) , and by (5), (14), (15), and (17) we have3(1 − d )(2 δ ′ − m ) n/m > sp( n, δ ) ≥ sc( n, δ ) and 3(1 − d ) n/ ≥ n/ > sp( n, δ ) ≥ sc( n, δ ). It follows that G contains both P n,δ ) and C ℓ foreach ℓ ∈ [3 , sc( n, δ )] \ { } as required. (cid:3) By (23), if there are two vertex disjoint edges in W , then we aredone by Fact 23. Thus we assume in the following that no such twoedges exist. This implies that there are two vertices in W which meetevery edge in W . Since neither of these two vertices has more than ξn neighbours in S I ⊆ W , while | I | > ( − η ) m by ( v ) and because δ < n/
3, there is a vertex in W adjacent to no vertex of W . Weconclude that(25) n − δ − ηn ≤ | [ I | ≤ | W | ≤ n − δ. Our next goal is to extract from each set S B i a large set A i ofvertices which are adjacent to almost all vertices in W and are such that G [ A i ] has minimum degree somewhat above | A i | /
2. Because at least | W | δ − | W | edges leave W , the total number of non-edges between W and V ( G ) \ W is at most | W || V ( G ) \ W | − | W | ( δ − ≤ ( n − δ )( δ + 11 ηn − δ + 2) (25) ≤ ηn + 2 n . In particular, by the definition of ξ , by (14) and (16),(26) (cid:12)(cid:12)(cid:12)(cid:8) v ∈ V ( G ) \ W : deg( v, W ) < | W | − ξ n (cid:9)(cid:12)(cid:12)(cid:12) ≤ ξ n . In addition, by assumption ( vi ) we have | B i | ≤ m (2 δ − n ) / (10 n ),which together with δ ≤ n/
3, (14), (15) and (22) implies(27) (cid:12)(cid:12)(cid:12) [ B i (cid:12)(cid:12)(cid:12) ≤ δ − n ) ≤ δ < δ − ξn − ( d + ε ) n . However, by assumption ( vi ) and the definition of an ( ε, d )-regularpartition, vertices in S B i send at most ( d + ε ) n edges to V ( G ) − S B i − S I . It follows from the definition of W that [ B i ∩ W = ∅ for all i ∈ [ k ] . Furthermore, (14), (15) and (22) imply that v ∈ S B i has at least(28) δ − | W | − ( d + ε ) n (25) ≥ δ − n − ( d + ε ) n (27) > | [ B i | / ξ n neighbours in S B i .Now, for each i ∈ [ k ] we let A i be the set of vertices in S B i whichare adjacent to at least | W |− ξ n vertices of W . In the rest of this para-graph we determine some important properties of the sets A i . By (26)we have(29) (cid:12)(cid:12)(cid:12) [ i ∈ [ k ] (cid:0) [ B i (cid:1) \ A i (cid:12)(cid:12) ≤ ξ n for all i ∈ [ k ] . ETWEEN TUR ´AN’S THEOREM AND P ´OSA’S CONJECTURE 33
But the vertices which are neither in W nor any of the sets A i must beeither in the exceptional set V or in S B i \ A i for some i . Hence wehave(30) (cid:12)(cid:12)(cid:12) V ∪ [ i ∈ [ k ] (cid:0) [ B i (cid:1) \ A i (cid:12)(cid:12)(cid:12) ≤ εn + ξ n < ξ n . Accordingly (28) implies that(31) δ ( G [ A i ]) ≥ | A i | / ξ n , and since | B i | > δ ′ − | I | ≥ δ ′ − m we have(32) | A i | ≥ | [ B i | − ξ n ≥ (1 − ε ) nm | B i | − ξ n ≥ δ − n − ξ n for each i ∈ [ k ], where we used the definition of ξ , (14), 15, and (18) inthe last inequality.In the remainder of the proof we utilize the sets A i in order to findthe desired squared path and squared cycles. We start by showing thatwe obtain squared cycles on ℓ vertices for each ℓ ∈ [3 , | A | ] \ { } . Tosee this note first that by Lemma 21 (with B = ∅ ) we find in A a copyof C ℓ ′ for each 2 ℓ ′ ∈ (cid:2) , min {| A | , n } (cid:3) . By the definition of A everyquadruple of consecutive vertices on such a cycle has at least | W |− ξ n common neighbours in W , and by the definition of ξ , (14), (15), and(25) we have | W | − ξ n ≥ n/
4. Hence we can apply Lemma 20 to G and W to square this cycle. This gives us squared cycles of lengths ℓ with 3 ≤ ℓ ≤ min n | A | , n o (17) = 32 | A | such that ℓ is divisible by three, but not of lengths not divisible bythree.If we seek a squared cycle C ℓ ′ +1 or C ℓ ′ +2 (with 3 ℓ ′ + 2 = 5) thenwe need to perform a process which we will call parity correction andwhich we explain in the following two paragraphs. We shall use thisparity correction process also in all remaining steps of the proof toobtain squared cycles of lengths not divisible by 3.For obtaining a squared cycle of length 3 ℓ ′ + 1 we proceed as fol-lows. We pick (using Tur´an’s theorem) a triangle abc in A and clone the vertex b , i.e., we insert a dummy vertex b ′ into G with the sameadjacencies as b . Then we apply Lemma 21 to A − { b } to find a path P = ( a, p , p , . . . , p ℓ ′ − , c ) on 2 ℓ ′ vertices whose end-vertices are a and c . Finally we apply Lemma 20 to the path bP b ′ , taking Q = ( b, a ), Q = ( b, a, p , p ) as the first quadruple and thereafter every other set offour consecutive vertices on P , finishing with ( p ℓ ′ − , p ℓ ′ − , c, b ′ ). Thisyields a squared path ( q , b, a, . . . , c, b ′ ) on 3( ℓ ′ +1) vertices, which givesa squared cycle ( b, a, . . . , c ) in G (without q and the clone vertex b ′ )on 3 ℓ ′ + 1 vertices as required. If we seek a squared cycle of length 3 ℓ ′ + 2 with ℓ ′ > A but two triangles abc , xyz connected with an edge cx (which we obtain by the Erd˝os-Stone theorem). We apply Lemma 21to find a path P = ( a, . . . , z ) in A \ { b, c, y, z } on 2 ℓ ′ vertices. Wethen apply Lemma 20 once to the path bP y and once to ( b, c, x, y ).Omitting the first vertex on each of the resulting squared paths andconcatenating, we get a squared cycle C ℓ ′ +2 .Hence we do indeed obtain squared cycles C ℓ for all ℓ ∈ [3 , | A | ] \{ } . It remains to show that we can also find C ℓ for all ℓ with | A | ≤ ℓ ≤ sc( n, δ ) and that we can find P n,δ ) .For this purpose, we first re-incorporate the vertices that are neitherin W nor in any of the sets A i by examining in which of the A i theyhave many neighbours. More precisely, for each i ∈ [ k ], we let X i be A i together with all vertices in V ( G ) \ W which are adjacent to atleast 30 ξ n vertices of A i . Because every vertex in V ( G ) \ W has atleast δ − | W | neighbours outside W , by (25) every vertex in G − W isin X i for at least one i . Moreover, by the definition of an ( ε, d )-regularpartition, assumption ( vi ) and since A j ⊆ S B j , we have for all j ∈ [ k ]with j = i that(33) A j ∩ X i = ∅ . Hence it follows from (30) that(34) | X i | < | A i | + 2 ξ n and | X − A | ≤ ξ n . We finish the proof by distinguishing three cases.
Case 1: | X i ∩ X j | ≥ i = j . Let v and v be distinct verticesof X i ∩ X j . Let u and u be distinct neighbours in A i of v and v respectively, and similarly y and y in A j . Applying Lemma 21 to A i we can find a path from u to u of length ℓ ′ for any 4 ≤ ℓ ′ ≤ | A i | − A j from y to y . Concatenating thesepaths with v and v we can find a 2 ℓ ′ -vertex cycle T ℓ ′ in X ∪ X forany 10 ≤ ℓ ′ ≤ | A i | + | A j | −
2. There are no quadruples of consecutivevertices on T ℓ ′ using both v and v . The four quadruples that useeither v or v each have at least ( ξ − ξ ) n > k common neighboursin W , where the inequality follows from (16), (22), from(35) k ≤ ν − , and from ξ − ξ >
0. All other quadruples have at least | W | − ξ n common neighbours in W . So applying Lemma 20 we obtain a squaredcycle on 3 ℓ ′ vertices. Again it is possible to perform parity corrections(prior to applying Lemma 21) so that in this case we have C ℓ ⊆ G forevery ℓ ∈ [3 , ( | A i | + | A j | − \ { } . By (32), we have sc( n, δ ) ≤ sp( n, δ ) < ( | A i | + | A j | − ETWEEN TUR ´AN’S THEOREM AND P ´OSA’S CONJECTURE 35
Case 2: for some i every vertex of A i is adjacent to at least onevertex outside X i ∪ W . Since | A i | (29) ≥ (cid:12)(cid:12)(cid:12) [ B i (cid:12)(cid:12)(cid:12) − ξ n (20) ≥ ν (1 − ε ) n − ξ n (22) ≥ ξn (22),(35) > kξ n we can certainly find j = i such that there are 31 ξ n vertices in A i all adjacent to vertices of X j \ X i . Since no vertex of X j \ X i is adja-cent to 30 ξ n vertices of A i (by definition of X i ), we find two disjointedges u v and u v from u , u ∈ A i to v , v ∈ X j . Choosing distinctneighbours y of v and y of v in A j and applying the same reasoningas in the previous case we are done. Case 3: for each i = j we have | X i ∩ X j | ≤
1, and for each i somevertex in A i is adjacent only to vertices in W ∪ X i . Thus | X i | ≥ δ − | W | + 1 for each i . We now first focus on finding a squared pathon sp( n, δ ) vertices in G , and then turn to the squared cycles whichwill complete the proof. If for some i = j we have | X i ∩ X j | = 1 thenwe obtain a squared path of the desired length as in Case 1. There werequired two vertices in X i ∩ X j to obtain a squared cycle (which mustreturn to its start), but one vertex suffices for a squared path to crossfrom X i to X j .So, assume that the sets X i are all disjoint. It follows that k ≤ ( n − | W | ) / ( δ − | W | + 1). Since | W | ≤ n − δ by (25), this implies k ≤ n − ( n − δ ) δ − ( n − δ ) + 1 = δ δ − n + 1 . Now if k ≥ r p ( n, δ ) + 1 then we would have r p ( n, δ ) + 1 ≤ k ≤ δ δ − n + 1 , and so r p ( n, δ ) ≤ n − δ − δ − n + 1 , but by (3) we have r p ( n, δ ) ≥ n − δ δ − n +1 , so k ≤ r p ( n, δ ) . Thus the largest of the sets X i , say X , has at least(36) | X | ≥ n − | W | k (25) ≥ δk ≥ δr p ( n, δ )vertices.We now want to apply Lemma 21 with H = G [ X ] and ‘bad’ vertices B = X − A . Note that by (34) there are at most 2 ξ n vertices in B = X − A , and so we have | B | (34) ≤ ξ n (22) ≤ νδ (35) ≤ δ k ≤ | H | . Moreover, δ ( H ) = δ ( G [ X ]) ≥ ξ n by definition of X , and thereforeevery vertex in B has at least 30 ξ n ≥ · ξ n ≥ | B | neighbours in H .In addition, vertices v in H − B ⊆ A satisfydeg( v, X ) (31) ≥ | A | ξ n (34) > | X | ξ n = | H | ξ n (16) ≥ | H | | B | + 10 . Hence we can indeed apply Lemma 21, to obtain a path T coveringmin { X , n/ } vertices on which every quadruple of consecutive verticescontains at most one ‘bad’ vertex. Finally we want to apply Lemma 20to the graph G [ X ∪ W ] and the cycle T with the following ordering σ ofthe quadruples of consecutive vertices in T : σ is such that all quadru-ples containing vertices from B come first, followed (by an arbitraryordering of) all other quadruples. There are at most 2 · ξ n quadruplescontaining vertices from B = X − A , and by the definition of A andof W , each of them has at least ( ξ − ξ ) n ≥ ξ n common neighboursin W . All remaining quadruples have, by the definition of A , by (25)and since δ ≤ n/
3, at least | W | − ξ n ≥ n ≥ min {| X | , n } commonneighbours in W . Hence, we can indeed apply Lemma 20 to obtain asquared path on at least min {| X | , n/ } ≥ sp( n, δ ) vertices, wherethe inequality follows from the definition of sp( n, δ ), from (17), andfrom (36).At last, we show that we can find in G the desired long squaredcycles in Case 3. Assume first that there is a cycle of sets (relabellingthe indices if necessary) X , X , . . . , X s for some 3 ≤ s ≤ k such that X i ∩ X i +1 mod s = { v i } for each i , and the v i are all distinct, then foreach i we may choose neighbours u i ∈ A i and y i in A i +1 mod s of v i ,and we may insist that all these 3 s vertices are distinct. Similarly asbefore we can apply Lemma 21 to each G [ A i ] in turn and concatenatethe resulting paths, in order to find a cycle T ℓ ′ for every 8 s ≤ ℓ ′ ≤| A | + | A | on which there are no quadruples using more than one vertexoutside S i A i . Again (checking the conditions similarly as before) wemay apply Lemma 20 to T ℓ ′ to obtain a squared cycle on 3 ℓ vertices.Finally by performing parity corrections we obtain C ℓ for every ℓ ∈ [3 , ( | A | + | A | )] \ { } .If there exists no such cycle of sets, then P ki =1 | X i | ≤ n − | W | + k − | X i | ≥ δ − | W | + 1 for each i and | W | ≤ n − δ , itfollows from the definition of r c ( n, δ ) (by establishing a relation similarto (3)) that k ≤ r c ( n, δ ), and by averaging, that the largest of thesets X i , say X , contains at least 2 sc( n, δ ) / X to obtain a cycle T ℓ ′ for each 4 ≤ ℓ ′ ≤ | X | on which the ‘bad’ vertices from B = X − A are separated, and applyLemma 20 to it to obtain a squared cycle C ℓ ′ for each 6 ≤ ℓ ′ ≤ sc( n, δ ) ETWEEN TUR ´AN’S THEOREM AND P ´OSA’S CONJECTURE 37 as required. Again the parity correction procedure is applicable, so weget C ℓ for every ℓ ∈ [3 , sc( n, δ )] \ { } . (cid:3) Concluding remarks
The proof of Theorem 4.
Our results were most difficult to provefor δ ≈ n/
7. This is somewhat surprising given the experience fromthe partial and perfect packing results of Koml´os [9] and K¨uhn andOsthus [15]. In the setting of these results it becomes steadily moredifficult to prove packing results as the minimum degree of the graph(and hence the required size of a packing) increases, with perfect pack-ings as the most difficult case. Yet in our setting it is relatively easyto prove our results when the minimum degree condition is large. Thisdifference occurs because we have to embed triangle-connected graphs;as the minimum degree increases the possibilities for bad behaviourwhen forming triangle-connections are reduced.This is paralleled by the behaviour of K -free graphs: For any mini-mum degree δ ( G ) > v ( G ) / G is not K -free. However, if δ ( G ) > v ( G ) / K -free graph G is forced to be tripartite, while for smaller values of δ ( G )there exist more possibilities. Extremal graphs.
It is straightforward to check (from our proofs)that up to some trivial modifications the graphs G p ( n, δ ) and G c ( n, δ )are the only extremal graphs. We believe that the graph G p ( n, δ ) re-mains extremal for squared paths even when δ is not bounded awayfrom n/
2, although as noted in Section 1 the same is not true for G c ( n, δ )and squared cycles.However it is not the case that the only extremal graph excludingsome C ℓ of chromatic number four is K n − δ,n − δ, δ − n (cf. ( ii ) of our maintheorem, Theorem 4). Let us briefly explain this. Suppose ℓ is notdivisible by three. Since C ℓ has no independent set on more than ⌊ ℓ/ ⌋ vertices, whenever we remove an independent set from C ℓ we mustleave some three consecutive vertices, which form a triangle. Nowsuppose that we can find a graph H on δ vertices with minimum degree2 δ − n which is both triangle-free and contains no even cycle on morethan 2(2 δ − n ) vertices. Then the graph G obtained by adding anindependent set of size n − δ to H , all of whose vertices are adjacent toall of H , contains no squared cycle of length indivisible by three andno squared cycle with more than 3(2 δ − n ) vertices.To mention one possible H , take δ = n and let H be obtainedas follows. We take the disjoint union of three copies of K n/ ,n/ and fix a bipartition. Now we add three vertex disjoint edges withinone of the two partition classes, one between each copy of K n/ ,n/ .The resulting triangle-free graph has no even cycle leaving any copy of K n/ ,n/ . Hence all even cycles have at most n vertices. However, ithas odd cycles of up to n − Long squared cycles.
Theorem 5 ( ii ) states that if any of variousodd cycles are excluded from G we are guaranteed even cycles of everylength up to 2 δ ( G ), whereas the equivalent statement in our Theorem 4contains an error term. We believe this error term can be removed, butat the cost of significantly more technical work, requiring both a newversion of the stability lemma and new extremal results. Higher powers of paths and cycles.
We note that Theorem 2 hasa natural generalisation to higher powers of cycles, the so called P´osa-Seymour Conjecture. This conjecture was proved for all sufficientlylarge n by Koml´os, S´ark¨ozy and Szemer´edi [12]. We conjecture a natu-ral generalisation of Theorem 4 for higher powers of paths and cycles.Given k , n and δ , we construct an n -vertex graph G ( k ) p ( n, δ ) by par-titioning the vertices into an ‘interior’ set of ℓ = ( k − n − δ ) verticesupon which we place a complete balanced k − n − ℓ vertices upon which we place a disjoint union of ⌊ ( n − ℓ ) / ( δ − ℓ + 1) ⌋ almost-equal cliques. We then join every ‘interior’vertex to every ‘exterior’ vertex. We construct G ( k ) c ( n, δ ) similarly, per-mitting the cliques in the ‘exterior’ vertices to overlap in cut-verticesof the ‘exterior’ set if this reduces the size of the largest clique whilepreserving the minimum degree δ . Conjecture 24.
Given ν > and k there exists n such that whenever n ≥ n and G is an n -vertex graph with δ ( G ) = δ > k − k n + νn , thefollowing hold. ( i ) If P kℓ ⊆ G ( k ) p ( n, δ ) then P kℓ ⊆ G . ( ii ) If C k ( k +1) ℓ ⊆ G ( k ) c ( n, δ ) for some integer ℓ , then C k ( k +1) ℓ ⊆ G . ( iii ) If C kℓ ⊆ G ( k ) c ( n, δ ) with χ ( C kℓ ) = k + 2 and C kℓ G for someinteger ℓ , then C k ( k +1) ℓ ⊆ G for each integer ℓ < kδ − ( k − n − νn . It seems likely that again the νn error term in the last statement isnot required, but again (at least for powers of cycles) it is required inthe minimum degree condition. Acknowledgement
This project was started at DocCourse 2008, organised by the re-search training group Methods for Discrete Structures, Berlin. In par-ticular, we would like to thank Mihyun Kang and Mathias Schacht fororganising this nice event.
References
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ETWEEN TUR ´AN’S THEOREM AND P ´OSA’S CONJECTURE 39
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For the proof of Lemma 8 we apply the following version (which isa special case) of the Blow-up Lemma of Koml´os, S´ark¨ozy and Sze-mer´edi [11].
Lemma 25 (Blow-up Lemma [11]) . Given fixed c, d > , there exist ε > and n bl such that for any < ε < ε the following holds.Let H be any graph on at least n bl vertices with V ( H ) = V ˙ ∪ V ˙ ∪ V and | V i | ≥ | V ( H ) | , in which each bipartite graph H [ V i , V j ] is (3 ε, d ) -regular and furthermore δ V i ( V j ) ≥ d | V i | for each ≤ i, j ≤ .Let F be any subgraph of the complete tripartite graph with parts V , V and V such that the maximum degree of F is at most four.Assume further, that at most four vertices x i ( i ∈ [4] ) of F are endowedwith sets C x i ⊆ V j such that x i ∈ V j and | C x i | ≥ c | V j | Then there is an embedding ψ : V ( F ) → V ( H ) of F into H with ψ ( x i ) ∈ C x i for i ∈ [4] . We also say that the x i in Lemma 25 are image restricted to C x i . Proof of Lemma 8.
Given d , we let c = d /
4. Now Lemma 25 givesvalues ε > n bl . We choose ε el = min( ε , d / ε < ε el and m el we choose n el = max (cid:18) m el n bl , m ε (cid:19) . Let n ≥ n el , let G be an n -vertex graph, and let R ′ be an ( ε, d )-reducedgraph of G on m ≤ m el vertices.Fix a set T ′ = { T ′ , . . . , T ′ CTF( R ′ ) / } of vertex-disjoint triangles in atriangle component of R ′ covering CTF( R ′ ) vertices. For each triangle T ′ i = X ′ i, X ′ i, X ′ i, we may by regularity for each j ∈ [3] remove atmost ε | X ′ i,j | vertices from X ′ i,j to obtain a set X i,j such that each pair( X i,j , X i,k ) is not only (2 ε, d )-regular but also satisfies δ X i,k ( X i,j ) ≥ ( d − ε ) | X i,k | . We let R be the (2 ε, d )-reduced graph correspondingto the new vertex partition given by replacing each X ′ i,j with X i,j ;then every edge of R ′ carries over to R , and we let T be the set ofCTF( R ′ ) / R corresponding to T ′ . We set r = CTF( R ′ ) / W , . . . , W r − and W ′ in R . Next, for each ofthese triangle walks W = ( E , E , . . . ) we do the following. Let ⇀ U V be(a suitable) orientation of the first edge E of W . We shall constructa sequence Q ( W, ⇀ U V ) of vertices of R whose first two vertices are U and V , in that order, and which has the property that every vertex inthe sequence is adjacent to the two preceding vertices (as is the casefor a squared path). Then we use this sequence Q ( W, ⇀ U V ) to obtaina squared path in G following W , whose first two vertices are in U and V . Finally, connecting suitable paths appropriately will lead to aproof of ( i ), ( ii ), and ( iii ).We first construct the triangle walks W , . . . , W r − and W ′ . For each1 ≤ i ≤ r − W i be a fixed triangle walk in R whose first edgeis in T i and whose last is in T i +1 . We suppose (repeating edges in the ETWEEN TUR ´AN’S THEOREM AND P ´OSA’S CONJECTURE 41 triangle walk W i if necessary) that each triangle walk W i contains atleast ten edges, that the first edge of W i +1 is not the same as the lastedge of W i , and such that each walk with more than ten edges is ofminimal length. We have | W i | ≤ (cid:0) m (cid:1) for each i . Let W ′ be the trianglewalk obtained by concatenating W , . . . , W r − .Next, we describe how to construct the sequence Q ( W, ⇀ A B ) forany triangle walk W = ( E , E , . . . ) in R and any orientation ⇀ A B ofits first edge E . We construct Q ( W, ⇀ A B ) iteratively as follows. Let Q = ( A , B ). Now for each 2 ≤ i ≤ | W | successively, we define Q i as follows. The last two vertices A i − , B i − of Q i − are an orientationof E i − . If E i = A i − B i we create Q i by appending ( B i , A i − ) to Q i − ;if E i = B i − B i we append ( B i , A i − , B i − , B i ) to Q i − to create Q i . Ateach step the final two vertices of Q i are an orientation of E i . Further-more every vertex of Q i is adjacent in R to the two vertices precedingit in Q i . Finally, we let Q ( W, ⇀ A B ) = Q | W | .We shall need the following observations concerning the lengths ofsequences constructed in this way. It is easy to check by inductionthat for any triangle walk W with at least two edges whose first edgeis U V , we have(37) | Q ( W, ⇀ U V ) | + | Q ( W, ⇀ V U ) | ≡ . Now consider the concatenation W ′ of the walks W i . Let U V be thefirst edge of W . If we construct Q ( W ′ , ⇀ U V ) then the first edge U i V i and the last edge U ′ i V ′ i of each W i obtains an orientation, say ⇀ U i V i and ⇀ U ′ i V ′ i . Clearly, there are sequences ˜ Q i of vertices in T i for 1 < i < r ,such that Q ( W ′ , ⇀ V U ) is the concatenation of Q ( W , ⇀ V U ) , ˜ Q , Q ( W , ⇀ V U ) , . . . , ˜ Q r − , Q ( W r − , ⇀ V r − U r − ) . Further we let ˜ Q = T − U V and ˜ Q r = T r − U ′ r − V ′ r − . We define f i = | ˜ Q i | mod 3 for i ∈ [ r ]. Together with (37) we obtain | Q ( W ′ , ⇀ U V ) | + | Q ( W , ⇀ V U ) | + X
Let X , X , X be vertices of R (not necessarily distinct),and Z be any set of at most ε | X | vertices of G . Suppose that X X and X X are edges of R . Suppose furthermore that we have two ver-tices u and v of G such that u and v have at least ( d − ε ) | X | commonneighbours in X , and v has at least ( d − ε ) | X | neighbours in X .Then there is a vertex w ∈ X − Z adjacent to u and v such that v and w have at least ( d − ε ) | X | common neighbours in X and w hasat least ( d − ε ) | X | neighbours in X .Proof of Fact 26. Let W be the set of common neighbours of u and v in X . Since X X ∈ E ( R ), at most 2 ε | X | vertices of W have fewerthan ( d − ε ) | Γ( v ) ∩ X | ≥ ( d − ε ) | X | common neighbours with v in X . Since X X ∈ E ( R ) at most 2 ε | X | vertices of W have fewerthan ( d − ε ) neighbours in X . Finally since 6 ε | X | < ( d − ε ) | X | we can find a vertex of W \ Z satisfying the desired properties. (cid:3) With these buiding bricks at hand we can now turn to the proofs of( i ), ( ii ), and ( iii ). Proof of ( i ), i.e., G contains C ℓ for each 3 ℓ ≤ (1 − d ) CTF( R ) n/m :When ℓ ≤ (1 − d ) n/m we have C ℓ ⊆ K (1 − d ) n/m, (1 − d ) n/m, (1 − d ) n/m andthus by Lemma 25 we can find C ℓ as a subgraph of G (whose verticesare in T , with no restrictions required). Otherwise we use the followingstrategy. Let U V be the first edge of the triangle walk W .Our first goal will be to construct a squared path P ′ in G which‘connects’ T to T , T to T , and so on. For this purpose we choosetwo adjacent vertices u and v of G in U and V respectively, such that u and v have ( d − ε ) n/m common neighbours in both the third vertexof T and the third vertex of Q ( W ′ , ⇀ U V ), such that v has ( d − ε ) n/m neighbours in the fourth vertex of Q ( W ′ , ⇀ U V ), and such that u has( d − ε ) n/m neighbours in V . This is possible by the regularity of thevarious pairs. (Observe that the required sizes for the neighbourhoodsand joint neighbourhoods are chosen large enough for an applicationof Lemma 25 in the triangle T .) Now we apply Fact 26 with thevertices u and v and the third, fourth and fifth vertices of Q ( W ′ , ⇀ U V )to obtain a third vertex v ′ in the third vertex of Q ( W ′ , ⇀ U V ) such that u and v are adjacent to v ′ . By repeatedly applying Fact 26 we construct asequence of vertices P ′ (starting with u, v, v ′ ), where the i th vertex of P ′ is in the i th set of Q ( W ′ , ⇀ U V ) and is adjacent to its two predecessors,and where the vertices are all distinct (noting that 3 | W ′ | < εn/m ).Thus P ′ is a squared path running from T to T r − following all thetriangle walks W i .In addition we construct similarly (and without re-using vertices) foreach 1 ≤ i ≤ r − P i following the triangle walk W i .However, this time we use the opposite orientation for the first edge:that is, instead of constructing P from Q ( W , ⇀ U V ) we use Q ( W , ⇀ V U ),and similarly for each P i we use the opposite orientation of the firstedge of W i to that used in P ′ . Again, for each P i we insist that the ETWEEN TUR ´AN’S THEOREM AND P ´OSA’S CONJECTURE 43 first two vertices have suitable neighbourhoods in T i , and the last twoin T i +1 , for an application of Lemma 25 in these triangles. Again, thisis possible by regularity.We note that the total number of vertices on all of these squaredpaths is not more than 6 m (cid:0) m (cid:1) < εn/m . Finally, we remove from T all vertices of P = P ′ ∪ P ∪ · · · ∪ P r − . Since at most εn/m verticesare removed, and each cluster of T has size at least (1 − ε ) n/m , evenafter removal all three pairs remain (3 ε, d )-regular and each cluster stillhas size at least (1 − ε ) n/m .Thus the conditions of Lemma 25 are satisfied, and hence we mayembed a squared path S into T , with the four restrictions that its firstvertex is a common neighbour of the first two vertices of P ′ , its seconda neighbour of the first vertex of P ′ , its penultimate vertex a neighbourof the first vertex of P and its final vertex a common neighbour of thefirst two vertices of P (noting that by choice of the first two verticesof P ′ and of P the sets to which these vertices are restricted are indeedof size cn/m because c = d / ℓ + f vertices for any integer ℓ ∈ [10 , (1 − d ) n/m ] (since3 · ε < d ), where f ∈ { , , } is as defined above (38). Similarlywe may apply Lemma 25 to each T i (2 ≤ i ≤ r ), after removing P from T i , to obtain squared paths S i of length 3 ℓ i + f i for any integer ℓ i ∈ [10 , (1 − d ) n/m ].Finally S = P ′ ∪ S ∪ P ∪ . . . ∪ P r − ∪ S r forms a squared cyclein G . It follows from (38) that the length of S is divisible by three.We conclude that indeed S = C k , where we may choose any integer k with 3 k ∈ [6 m , (1 − d ) CTF( R ) n/m ], as required. Proof of ( ii ) : When every triangle component of R contains K we also want to obtain squared cycles whose lengths are not divisibleby three. Observe that if ABCD is a copy of K in R , then the ver-tex sequences ABC , ABCDABC and
ABCDABCDABC each startand end with the same pair. Hence, with the help of Fact 26, these se-quences can be used to construct squared paths in G of length 3 (whichis 0 mod 3), length 7 (1 mod 3), and length 11 (2 mod 3).We construct C ℓ for ℓ ∈ [3 , \ { } within a copy of K in R directly (by the above methods). To obtain C ℓ with 21 ≤ ℓ ≤ − d ) n/m we remove at most 2 εn/m vertices from each of A , B and C to obtain a triangle satisfying the conditions of Lemma 25, construct ashort path in A, B, C, D following the appropriate vertex sequence for ℓ mod 3 and apply Lemma 25 to obtain C ℓ . Finally, to obtain longersquared cycles we perform the same construction as above, with theexception that W ′ is any triangle walk to and from a copy of K , andso Q ( W ′ , ⇀ U V ) may be taken (using one of the three vertex sequencesabove) to have any desired number of vertices modulo three (and notmore than 2 m in total). Proof of ( iii ) : Lastly, when we are required to construct a squaredpath between two specified edges u v (with 2 dn/m common neigh-bours in both X and Y ) and u v (with 2 dn/m common neighboursin both X and Y ) using triangles T in R , we apply the identical strat-egy, noting that the conditions on u v and u v are already suitablefor an application of Fact 26.are already suitablefor an application of Fact 26.