Color-Critical Graphs Have Logarithmic Circumference
aa r X i v : . [ m a t h . C O ] A p r Color-Critical Graphs Have Logarithmic Circumference
Asaf Shapira ∗ Robin Thomas † Abstract
A graph G is k -critical if every proper subgraph of G is ( k − G itself is not. We prove that every k -critical graph on n vertices has a cycle of length at leastlog n/ (100 log k ), improving a bound of Alon, Krivelevich and Seymour from 2000. Examples ofGallai from 1963 show that the bound cannot be improved to exceed 2( k −
1) log n/ log( k − k -critical graphs, raised byDirac in 1952 and Kelly and Kelly in 1954. All graphs in this paper are finite and simple; that is, they have no loops or multiple edges. Pathsand cycles have no “repeated” vertices. A graph G is k -critical , where k ≥ G is ( k − G itself is not. There is an easy descriptionof k -critical graphs for k ≤
3, but for k ≥ k -critical graphs was introduced in the 1940s by Dirac as part of his PhD Thesis.Since then k -critical graphs have been studied extensively, as documented for instance in [8, Chap-ter 5]. In this paper we study the circumference of k -critical graphs, where the circumference ofa graph G is the length of the longest cycle in G . The only 3-critical graphs are odd cycles, butfor k ≥ k ≥ n > k + 1 let L k ( n ) denote the largest integer l such that every k -critical graph on n vertices has circumference atleast l . Elementary constructions show that the function L k ( n ) is well-defined for all integers k ≥ n > k + 1. The study of the function L k ( n ) originated in the work of Dirac [5] and Kelly andKelly [9].As every k -critical graph has minimum degree at least k −
1, we have L k ( n ) ≥ k . Dirac [5] showedthat L k ( n ) ≥ k − n ≥ k − k -critical graphs should contain much ∗ School of Mathematics and School of Computer Science, Georgia Institute of Technology, Atlanta, GA 30332.Supported in part by NSF Grant DMS-0901355. † School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0160. Supported in part by NSF Grantnumber DMS-0739366. 21 August 2009. k we have lim n →∞ L k ( n ) = ∞ andthat actually L k ( n ) ≥ c √ n .The first non-trivial bounds on L k ( n ) where obtained in 1954 by Kelly and Kelly [9] who showedthat lim n →∞ L k ( n ) = ∞ , (1)thus confirming the first conjecture of Dirac mentioned above. According to [1] Kelly and Kelly [9]actually proved that L k ( n ) ≥ s log log n log log log n (2)for every fixed k ≥ n . They also showed thatlim inf n →∞ L ( n ) / log n ≤ / log (27 / , (3)thus disproving Dirac’s second conjecture for k = 4. Dirac [6] later extended the upper bound of [9]to all k ≥
4, and Read [12] later improved the upper bound by showing thatlim inf n →∞ L k ( n ) / (log n · log (2) n · log (3) n · · · log ( k − n · (log ( k − n ) ) ≤ (2 / log 4) k − , (4)where log ( i ) ( x ) is the i -times iterated logarithm function. The best known upper bound on L k ( n ) wasobtained in 1963 by Gallai [7], who improved and significantly simplified the previous constructionsby showing that for every integer k ≥ n such that L k ( n ) ≤ k − k −
2) log n . (5)We present Gallai’s examples in Section 6, and we also point out that the same graphs establish therelated fact that for every integer k ≥ n such thatthere exists a k -critical graph on n vertices with no path of length exceeding 4( k − k −
2) log n . (6)As for lower bounds on L k ( n ), the first improvement of the result of Kelly and Kelly [9] cameafter almost 50 years, when Alon, Krivelevich and Seymour [1] obtained the following (exponential)improvement of (2) for all integers k ≥ n ≥ k + 2: L k ( n ) ≥ s log( n − k − . (7)The proof of (7) in [1] is based on a result (implicit in [1]) which says that every k -critical graphon n vertices has a path of length at least log n/ (log( k − heorem 1.1 For every integer k ≥ and every integer n ≥ k + 2 we have L k ( n ) ≥ log n
100 log k .
The following corollary solves, for every fixed k ≥
4, the problem of determining the order ofmagnitude of L k ( n ). The problem originated in the work of Dirac [5] and Kelly and Kelly [9],and is also stated in [8, Problem 5.11]. The lower bound follows immediately from Theorem 1.1; theupper bound follows by a minor modification of Gallai’s proof of (5) and is presented as Theorem 6.1. Corollary 1.2
For every integer k ≥ and every integer n ≥ k + 2log n
100 log k ≤ L k ( n ) ≤ k − k −
2) log n + 2 k . The corollary raises the obvious question whether there exist a function f and absolute constants c , c such that c f ( k ) log n ≤ L k ( n ) ≤ c f ( k ) log n . This remains an interesting open problem. Therelated (and perhaps easier) question, where we ask for the length of the longest path, is also open.Currently, the best known bounds for the latter problem are given by (6) and Lemma 2.2.There is a related problem, formulated by Neˇsetˇril and R¨odl at the International Colloquium onFinite and Infinite Sets in Keszthely, Hungary in 1973; see [10] for a detailed history of the problem.Neˇsetˇril and R¨odl asked whether it is true that for every two integers k, n ≥ N such that every k -critical graph on at least N vertices has a ( k − n vertices. For k = 4 the answer is yes, for the following reason. By (1) a large enough k -criticalgraph G has a long cycle C . Since G is not bipartite, it has an odd cycle, say C ′ . The graph G is2-connected by Lemma 3.2(i) below. Now an elementary argument using just the 2-connectivity of G shows that G has an odd cycle of length at least | V ( C ) | /
2. (The details may be found in [1].)This argument and Theorem 1.1 imply the following corollary.
Corollary 1.3
Let k, n ≥ be integers. Then every k -critical graph on n vertices has a -criticalsubgraph on at least log n/ (200 log k ) vertices. This is an improvement over the bound p log n/ log( k −
1) of Alon, Krivelevich and Seymour [1].The problem of Neˇsetˇril and R¨odl is open for all k ≥ k -critical graphs which will be used in the proof of Theorem 1.1. InSection 4 we prove a variation of a theorem of Bondy and Locke [2], stated as Theorem 2.4 below.The proof of Theorem 1.1 appears in Section 5. In Section 6 we present Gallai’s construction thatleads to the upper bound (5) and statement (6), and point out how to deduce the upper bound inCorollary 1.2 from it. 3 Proof Overview If T is a tree and x, y ∈ V ( T ), then there is a unique path in T with ends x and y , and we will denoteit by xT y . Let G be a graph, let T be a spanning tree of G , and let v ∈ V ( T ). The tree T is calleda depth-first search (DFS) spanning tree rooted at v if for every edge xy ∈ E ( G ) either x ∈ V ( vT y )or y ∈ V ( vT x ). It is easy to see that for every connected graph G and every vertex v ∈ V ( G ) thereis a DFS spanning tree in G rooted at v . For X = ∅ the following lemma is implicit in [1]. Thestraightforward generalization will be needed later in the paper. Lemma 2.1
Let G be a k -critical graph on n vertices, let X ⊆ V ( G ) have size s , and let T be aDFS spanning tree of G \ X rooted at a vertex t . Then for every integer j ≥ the number of verticesof T at distance exactly j from t is at most ( k − s if j = 1 and ( k − j − ( k − s otherwise. Proof.
Let
G, X, T, t be as stated, let X = { x , x , . . . , x s } , and let t ∈ V ( T ) be a vertex atdistance j ≥ t in T . We wish to define a sequence Q ( t ). Let t , t , . . . , t j − , t j = t be thevertices of the path tT t , listed in order. Since G is k -critical, the graph G \ tt j − obtained from G bydeleting the edge tt j − is ( k − t is adjacent to t it has a ( k − φ suchthat φ ( t ) = 1 and φ ( t ) = 2. We define Q ( t ) := ( φ ( t ) , φ ( t ) , . . . , φ ( t j − ) , φ ( x ) , φ ( x ) , . . . , φ ( x s )).Since for i = 1 , , . . . , j the vertex t i is adjacent to t i − , there are at most ( k − j − ( k − s sequencesthat arise this way (at most ( k − s if j = 1). It follows that if there are more than ( k − j − ( k − s vertices at distance j from t in T (or more than ( k − s if j = 1), then there are two vertices t, t ′ at distance j from t such that Q ( t ) = Q ( t ′ ). Let t ′ = t , t ′ , . . . , t ′ j − , t ′ j = t ′ be the vertices of thepath t T t ′ , let p be the largest integer such that t p = t ′ p , and let the set Z consist of the vertex t p +1 and all its descendants in the rooted tree ( T, t ). Let φ be a coloring as above, and let φ ′ be theanalogous coloring of G \ t ′ t ′ j − . The fact that Q ( t ) = Q ( t ′ ) implies that φ ( u ) = φ ′ ( u ) for every u ∈ X ∪ V ( t T t p ) . (8)We now define a coloring ψ by ψ ( u ) = φ ( u ) for every u ∈ V ( G ) − Z and ψ ( u ) = φ ′ ( u ) for every u ∈ Z . Since T is a DFS spanning tree of G \ X , it follows that every edge of G with one end in Z and the other end in V ( G ) − Z has the other end in X ∪ V ( t T t p ). It follows from (8) that ψ is avalid ( k − G , contrary to the k -criticality of G .The above lemma is the main tool in the proof of (7). Alon, Krivelevich and Seymour [1] use itto deduce that every k -critical graph has a long path, as follows. Lemma 2.2
For every integer k ≥ every k -critical graph on n vertices has a path of length at least log n/ log( k − . roof. Let k ≥ G be a k -critical graph on n vertices, and let v ∈ V ( G ) be somevertex in G . Since G is connected, it has a DFS spanning tree T rooted at v . Let h be the length ofa longest path in T with one end v . By Lemma 2.1 applied with X = ∅ we deduce that n ≤ k −
2) + ( k − + · · · + ( k − h − ≤ h − X j =0 ( k − j ≤ ( k − h , because h ≥ k ≥
4. It follows that h ≥ log n/ log( k − Lemma 2.3
If a -connected graph has a path of length l , then it has a cycle of length at least √ l . Since every k -critical graph is 2-connected by Lemma 3.2 below, the lower bound (7) of Alon,Krivelevich and Seymour follows immediately from Lemmas 2.2 and 2.3. However, the bound inLemma 2.3 is tight and the bound in Lemma 2.2 is asymptotically tight by (6), and so an improvementof (7) requires a different strategy. For 3-connected graphs the bound can be dramatically improved,as shown by Bondy and Locke [2]: Theorem 2.4
If a -connected graph has a path of length l then it has a cycle of length at least l/ . So combining Lemma 2.2 and Theorem 2.4 we get that every 3-connected k -critical graph has acycle of length at least 2 log n/ (5 log k ). Unfortunately, not all k -critical graphs are 3-connected, butthose that are not can be constructed from two smaller k -critical graphs. This is a result of Dirac [4]and is described in Lemma 3.2; a proof may also be found in [11, Problem 9.22]. Thus one mighthope that we could use this result of Dirac and apply induction. That is indeed our strategy, but itturns out that it is not enough (at least in our proof) to just decompose the graph once and applyinduction; instead, we need to break the graph repeatedly by (non-crossing) cutsets of size two anduse the resulting tree-structure. That brings us to the notion of tree-decomposition, which formalizesthis break up process. Definition 2.5 A tree decomposition of a graph G is a pair ( T, W ) where T is a tree and W =( W t : t ∈ V ( T )) is a collection of subsets of V ( G ) such that • S t ∈ V ( T ) W t = V ( G ) and every edge of G has both ends in some W t , and • If t, t ′ , t ′′ ∈ V ( T ) and t ′ belongs to the unique path in T connecting t and t ′′ , then W t ∩ W t ′′ ⊆ W t ′ .For t ∈ V ( T ) we define the torso of ( G, T, W ) at t to be the graph with vertex-set W t in which u, v ∈ W t are adjacent if either they are adjacent in G or u, v ∈ W t ′ for some neighbor t ′ of t in T .We say that the tree-decomposition ( T, W ) is standard if | W t ∩ W t ′ | = 2 for every edge tt ′ ∈ E ( T )5nd each torso of ( G, T, W ) is 3-connected or a cycle. (A graph G is t -connected if it has at least t + 1 vertices and G \ X is connected for every set X ⊆ V ( G ) of size at most t − k -critical graph has a standard tree-decomposition. Thetorsos are not necessarily critical, but they are very close, so that is not really an issue, and so forthe purpose of this outline we can pretend that every torso satisfies the conclusion of Lemma 2.2.By Theorem 2.4 we deduce that each torso has a sufficiently long cycle, but we need more. Weactually need a “linkage”, a set of two disjoint paths with prescribed ends so that we can combinethese linkages in individual torsos to produce a cycle in the original graph. We deduce the existenceof such a linkage of desired length from Theorem 2.4 in Lemma 4.1, but only under the assumptionthat the two sets of prescribed ends are disjoint from each other; otherwise Lemma 4.1 is false. Thuswhen the two sets of prescribed ends are not disjoint we need a different method. In that case weare really looking for one path rather than a linkage, and we use Lemma 2.1 to find it. In this section we prove some preliminary lemmas and introduce several concepts that will be usedlater on in the proof of Theorem 1.1 in Section 5. Our first lemma is well-known and appears in [3,Exercise 12.20].
Lemma 3.1
Every -connected graph has a standard tree-decomposition. Let G be a graph, and let x, y be distinct vertices of G . If x, y are not adjacent, then we define G + xy to be the graph obtained from G by adding an edge joining x and y , and if x, y are adjacent,then we define G + xy to be G . We define G/xy to be the graph obtained from G by deleting the edge xy (if it exists), identifying the vertices x and y and deleting all resulting parallel edges. Actually,in all applications of the operation G/xy in this paper the vertices x and y will not be adjacent, andthey will have no common neighbors, so the clause about deleting parallel edges will not be needed.Statement (iv) of the following lemma is due to Dirac [4], and a proof may also be found in [11,Problem 9.22]. Lemma 3.2
Let k ≥ be an integer, let G be a k -critical graph, and let u, v ∈ V ( G ) be such that G \{ u, v } , the graph obtained from G by deleting the vertices u and v , is disconnected. Then (i) u = v , and hence G is -connected, (ii) u is not adjacent to v , (iii) G \{ u, v } has exactly two components, and (iv) there are unique proper induced subgraphs G , G of G such that G = G ∪ G , V ( G ) ∩ V ( G ) = { u, v } , the graphs G \{ u, v } and G \{ u, v } are the two components of G \{ u, v } , u, v haveno common neighbor in G , and G + uv and G /uv are k -critical. roof. Since G \{ u, v } is disconnected, there is an integer s ≥ G can beexpressed as G = G ∪ G ∪ · · · ∪ G s , where the graphs G i are pairwise edge-disjoint, each has at leastone edge, and V ( G i ∩ G j ) = { u, v } for distinct integers i, j ∈ { , , . . . , s } . Since G is k -critical, each G i is ( k − u = v , then each G i has a ( k − u color 1.Those colorings can be combined to produce a ( k − G , contrary to the k -criticality of G . Thus u = v , and statement (i) follows.We now prove (ii) and (iii) simultaneously. If one of them does not hold, then we may assumethat s = 3. (If (ii) does not hold, then G may be chosen to consist of u , v , and the edge joiningthem.) Let us say that G i is of type one if some ( k − G i gives u and v the same color,and let us say it is of type two if some ( k − G i gives u and v different colors. Thus each G i is of type one or type two. We claim that no G i is of both types. For suppose for a contradictionthat G is of both types. But G and G are of the same type, because G ∪ G is ( k − k -criticality of G , and hence it follows that G ∪ G ∪ G = G is ( k − G i is of both types. We may assume that G and G are of the same type and that G isof different type. But then G ∪ G is not ( k − k -criticality of G . Thisproves (ii) and (iii). In particular, s = 2.We now prove (iv). We may assume from the symmetry that G is of type one and G is of typetwo. Since G is not k -colorable, it follows that G is not of type two and G is not of type one. Itfollows that neither G + uv nor G /uv is ( k − u, v have nocommon neighbor in G and that every proper subgraph of G + uv and G /uv is ( k − w ∈ V ( G ) − { u, v } is a neighbor of both u and v in G . Then the graph G \ uw has a ( k − φ by the k -criticality of G . Since G is not of type two we deduce that φ ( u ) = φ ( v ). But φ ( v ) = φ ( w ),because v is adjacent to w in G \ uw . Thus φ is ( k − G , a contradiction. This provesthat u, v have no common neighbor in G . Let e be an edge of G + uv . If e = uv , then let ψ bea ( k − G \ e . We have ψ ( u ) = ψ ( v ), because G is not of type one, and hence ψ is a( k − G + uv ) \ e . For e = uv we note that G is ( k − k -criticality of G . Finally, let f be an edge of G /uv , and let λ be a ( k − G \ f . Since G is not of typetwo, λ ( u ) = λ ( v ), and hence λ can be converted to a ( k − G /uv ) \ f , as desired.The first three statements of Lemma 3.2 have the following consequence. Lemma 3.3
Let k ≥ be an integer, let G be a k -critical graph, and let ( T, W ) be a standardtree-decomposition of G . (i) If t , t are adjacent in T , then the two vertices in the set W t ∩ W t are not adjacent, and (ii) if t , t are distinct neighbors of t in T , then W t ∩ W t = W t ∩ W t . roof. We prove only (ii), leaving (i) to the reader. Suppose for a contradiction that W t ∩ W t = W t ∩ W t , and let X denote this 2-element set. Let i ∈ { , , } . Since ( T, W ) is standard, W t i hasat least three elements, and hence there exists a vertex v i ∈ W t i − X . Then v , v , v belong to threedifferent components of G \ X , contrary to Lemma 3.2(iii).Part (iv) of Lemma 3.2 leads to the following construction, which will modify each torso of atree-decomposition of a k -critical graph and turn it into a k -critical graph. Let G be a k -criticalgraph and let ( T, W ) be a standard tree-decomposition of G such that each W t has at least threeelements. Let t ∈ V ( T ), and let u, v ∈ W t be distinct. We say that the pair uv is a virtual edge of W t if W t ∩ W t ′ = { u, v } for some neighbor t ′ of t in T . Thus Lemma 3.3 asserts that the virtual edges ofeach W t are pairwise distinct, and that they are not edges of G (but they are edges of the torso at t ,by definition of torso). We now classify virtual edges of W t into additive and contractive, as follows.Let uv be a virtual edge of W t , and let t ′ be the neighbor of t in T such that W t ∩ W t ′ = { u, v } .Since G \{ u, v } is disconnected, there exist graphs G , G as in Lemma 3.2(iv). Then W t is a subsetof exactly one of V ( G ) , V ( G ); if W t ⊆ V ( G ), then we say that the virtual edge uv is additive ;otherwise we say that it is contractive . We now define a graph N t as the graph obtained from G [ W t ]by adding the edge uv for every additive virtual edge uv of W t , and identifying the vertices u and v for every contractive virtual edge uv of W t . In other words, N t can be regarded as being obtainedfrom the torso of ( G, T, W ) at t by contracting all contractive virtual edges of W t . We call N t the nucleus of ( G, T, W ) at t . The next lemma shows that the nucleus is well-defined in the sense thatthe vertex identifications used during the construction do not produce loops or parallel edges. Lemma 3.4
Let k ≥ be an integer, let G be a k -critical graph, let ( T, W ) be a standard tree-decomposition of G , let t ∈ V ( T ) and let H denote the torso of ( G, T, W ) at t . Then (i) the subgraph of H induced by contractive virtual edges of W t is a forest, and for every com-ponent R of this forest and every v ∈ V ( H ) − V ( R ) , at most one vertex of R is adjacent to v in H ,and (ii) the nucleus N of ( G, T, W ) at t is k -critical. Proof.
We proceed by induction on the number of vertices of T . If T has only one vertex, thenthere are no virtual edges and N t = G , and hence both statements of the lemma hold. We maytherefore assume that T has more than one vertex, and that the lemma holds for all k -critical graphsthat have a standard tree-decomposition using a tree with strictly fewer than | V ( T ) | vertices. Let t ′ be a neighbor of t in T , and let W t ∩ W t ′ = { u, v } , so that uv is a virtual edge of W t . Let T ′ bethe component of T \ tt ′ containing t , let W ′ = ( W r : r ∈ V ( T ′ )), and let G ′ be the subgraph of G induced by the union of all W r over all r ∈ V ( T ′ ).Assume first that uv is an additive virtual edge. Then G ′ + uv is k -critical by Lemma 3.2(iv)and ( T ′ , W ′ ) is a standard tree-decomposition of G ′ + uv , where T ′ has strictly fewer vertices than8 . Furthermore, H is equal to the torso of ( G ′ + uv, T ′ , W ′ ) at t , and N is equal to the nucleusof ( G ′ + uv, T ′ , W ′ ) at t . Thus both conclusions follow by induction applied to G ′ + uv and thetree-decomposition ( T ′ , W ′ ). This completes the case when uv is an additive virtual edge.We may therefore assume that uv is a contractive virtual edge. In this case we proceed analo-gously, applying induction to the graph G ′ /uv and the tree-decomposition obtained from ( T ′ , W ′ ) byreplacing each occurrence of u or v by the new vertex of G ′ /uv that resulted from the identificationof u and v . In the proof of (i) we take advantage of the provision in Lemma 3.2(iv) that guaranteesthat u, v have no common neighbor in G ′ . Lemma 3.5
If a graph N is obtained from a graph H by repeatedly contracting edges, each timecontracting an edge that belongs to no triangle, and H has minimum degree at least three, then | E ( H ) | ≤ | E ( N ) | . Proof.
Let d , d . . . , d n be the degree sequence of N , and let us consider the reverse process thatproduces H starting from N . Then the i th vertex of N gives rise to at most d i − H .Thus | E ( H ) | ≤ | E ( N ) | + ( d −
3) + ( d −
3) + ... + ( d n − ≤ | E ( N ) | . as desired. Lemma 3.6
Let k ≥ be an integer, let G be a k -critical graph, let ( T, W ) be a standard tree-decomposition of G , and let t ∈ V ( T ) . Then the torso of ( G, T, W ) at t is -connected. Proof.
Let H denote the torso of ( G, T, W ) at t . If H is not 3-connected, then it is a cycle by thedefinition of standard tree-decomposition. But then the nucleus of ( G, T, W ) at t is a cycle, becauseit is obtained from H by contracting edges, contrary to Lemma 3.4(ii). Lemma 3.7
Let k ≥ be an integer, let G be a k -critical graph, let ( T, W ) be a standard tree-decomposition of G , let t ∈ V ( T ) , and let N be the nucleus of ( G, T, W ) at t . Then deg T ( t ) ≤ | E ( N ) | . Proof.
Let H be the torso of ( G, T, W ) at t . We first notice that deg T ( t ) ≤ | E ( H ) | , because eachneighbor of t in T gives rise to a unique virtual edge of G at t by Lemma 3.3(ii), and each virtualedge belongs to H . The graph H is 3-connected by Lemma 3.6. By Lemma 3.4(i) the graph N isobtained from H as in Lemma 3.5, and hence | E ( H ) | ≤ | E ( N ) | by that lemma, as desired. Lemma 3.8
Let k ≥ be an integer, let G be a k -critical graph, let ( T, W ) be a standard tree-decomposition of G , and let t ∈ V ( T ) . Then the nucleus of ( G, T, W ) at t has at least as many edgesas G [ W t ] . roof. This follows from the fact that no edges of G [ W t ] are lost during the construction of thenucleus. Lemma 3.9
Let k ≥ be an integer, let G be a k -critical graph, let ( T, W ) be a standard tree-decomposition of G , let t ∈ V ( T ) , and let N be the nucleus of ( G, T, W ) at t . Then the torso of ( G, T, W ) at t has a path of length at least log | E ( N ) | / log k . Proof:
Let H denote the torso of ( G, T, W ) at t . By Lemma 3.4 the graph N is k -critical, andso by Lemma 2.2 it has a path of length at least log | V ( N ) | / log k ≥ log | E ( N ) | / log k . Since N isobtained from H by contracting edges, H has a path at least as long.Finally we need an easy lemma about trees. If G is a graph, φ : V ( G )
7→ { , , , , . . . } is amapping, and H is a subgraph of G , then we define φ ( H ) := P v ∈ V ( H ) φ ( v ). Lemma 3.10
Let k ≥ be an integer, let T be a tree, let r ∈ V ( T ) , and assume that for every integer l ≥ there are at most k l vertices at distance exactly l from r in T . Let φ : V ( T )
7→ { , , . . . } be aweight function with φ ( r ) = 0 and φ ( t ) = 0 for at least one vertex in t ∈ V ( T ) . Then there exists avertex t ∈ V ( T ) at distance exactly l from r in T such that φ ( t ) > and l log k + log φ ( t ) ≥ log φ ( T ) . Proof.
For every integer l ≥ D l be the set of vertices of T at distance exactly l from r . Since φ ( r ) = 0 we have that for some l ≥ X t ∈ D l φ ( t ) ≥ φ ( T ) / l , since if this is not the case, then φ ( T ) = X l ≥ X t ∈ D l φ ( t ) < φ ( T ) X l ≥ − l ≤ φ ( T ) , a contradiction. Since | D l | ≤ k l , we deduce that there is a vertex t ∈ D l satisfying φ ( t ) ≥ φ ( T ) / l k l ≥ φ ( T ) /k l , (9)because k ≥
2. It follows that2 l log k + log φ ( t ) ≥ l log k + log φ ( T ) − l log k = log φ ( T ) , as desired. 10 An Application of the Theorem of Bondy and Locke
Let G be a graph, and let X, Y ⊆ V ( G ) be disjoint sets of size two. A linkage in G from X to Y isa set { P , P } of two disjoint paths, each with one end in X and the other end in Y . The length ofthe linkage is defined to be | E ( P ) | + | E ( P ) | . The following is the main result of this section. Lemma 4.1
Let G be a -connected graph, let X, Y ⊆ V ( G ) be disjoint sets of size two, and supposethat G has a path of length at least l . Then G has a linkage from X to Y of length at least l/ . The assumption that the sets
X, Y be disjoint is necessary, as the following example shows. Let t ≥ H be the graph obtained from a path with vertex-set { v , v , . . . , v t } inorder by adding an edge joining v it and v ( i +2) t for every i = 0 , , . . . , t −
2. Finally, let G be obtainedfrom H by adding a vertex u joined to all vertices of H . One can easily verify that G is 3-connected,has a path of length t , and yet every linkage from { u, v } to { u, v } has length at most linear in t .In the proof of Lemma 4.1 we will make use of the following lemma, which follows from thestandard “augmenting path” proof of Menger’s theorem or the Max-Flow Min-Cut Theorem; see,for instance [3, Section 3.3]. Lemma 4.2
Let r ≥ be an integer, let G be an r -connected graph, let S and T be two subsetsof the vertex-set of G , each of size at least r , and let P , P , . . . , P r − be disjoint paths such thatfor i = 1 , , ..., r − , the path P i has ends s i ∈ S and t i ∈ T . Then there exist disjoint paths Q , Q , . . . , Q r in G between S and T in such a way that all but one of the paths Q i has an end in { s , s , ..., s r − } , and all but one of the paths Q i has an end in { t , t , ..., t r − } . The proof of Lemma 4.1 will consist of three steps. In the first step we will obtain either arequired linkage, or a similar structure we call hammock , which we introduce next. In the secondstep we show that if a 3-connected graph has a long hammock, then it has either a long linkage,or a long “non-singular” hammock. Finally, we show how to get a required long linkage from theexistence of a long non-singular hammock.A hammock in G from X to Y is a quadruple η = ( P , P , R , R ), where • { P , P } is a linkage from X to Y , where P i has ends x i ∈ X and y i ∈ Y , • R i is a path with ends s i ∈ V ( P ) and t i ∈ V ( P ), and is otherwise disjoint from P ∪ P , • the paths R , R are disjoint, except possibly s = s , • the vertices x , s , s , y occur on P in the order listed (but are not necessarily distinct), and • the vertices x , t , t , y occur on P in the order listed (but are not necessarily distinct).The length of the hammock η is defined to be | E ( R ) | . (It may seem more natural to define thelength of η to be | E ( R ) | + | E ( R ) | . Indeed, by doing so it is possible to improve the constant 25in Lemma 4.1 to 17 .
5, but only at the expense of more extensive case analysis. The extra effort didnot seem justified.) We say that η is singular if s = s , and non-singular otherwise.11et us recall that if P is a path and u, v ∈ V ( P ), then by uP v we denote the unique subpath of P with ends u and v . Lemma 4.3
Let G be a -connected graph, let X, Y ⊆ V ( G ) be disjoint sets of size two, and assumethat G has a cycle C of length l . Then G has either a linkage from X to Y of length at least l/ , ora hammock from X to Y or from Y to X of length at least l/ . Proof:
Assume first that there exist four disjoint paths P , P , P , P , each with one end in X ∪ Y and the other end in V ( C ). For i = 1 , , , u i and v i be the ends of P i such that u i ∈ V ( C ), v , v ∈ X and v , v ∈ Y . If u , u , u , u occur on C in the order listed, then let C denote thesubpath of C \{ u , u } with ends u and u , and let C , C , and C be defined analogously. Theneither P ∪ P ∪ P ∪ P ∪ C ∪ C or P ∪ P ∪ P ∪ P ∪ C ∪ C is a linkage from X to Y of lengthat least l/
2, as desired. Thus we may assume that u , u , u , u occur on C in the order listed. Usinganalogous notation, if | E ( C ) | + | E ( C ) | ≥ l/
5, then P ∪ P ∪ P ∪ P ∪ C ∪ C is a linkage from X to Y in G of length at least l/
5. Thus we may assume that | E ( C ) | + | E ( C ) | ≥ l/
5, and so fromthe symmetry we may assume that | E ( C ) | ≥ l/
5. Then ( P ∪ C ∪ P , P ∪ C ∪ P , C , C ) isa hammock from X to Y in G of length at least 2 l/
5, as desired. This completes the case when G has four disjoint paths from X ∪ Y to V ( C ).We may therefore assume that those four paths do not exist, and hence by Menger’s theorem G can be expressed as G ∪ G , where | V ( G ) ∩ V ( G ) | = 3, X ∪ Y ⊆ V ( G ) and V ( C ) ⊆ V ( G ).Since G is 3-connected there exist three disjoint paths P , P , P from X ∪ Y to V ( C ) with nointernal vertices in X ∪ Y ∪ V ( C ). By symmetry, we may assume that P i has ends u i and v i , where u i ∈ V ( C ), v , v ∈ X and v ∈ Y . Then for i = 1 , , V ( G ) ∩ V ( G ) ∩ V ( P i ) includes aunique vertex, say w i . Thus V ( G ) ∩ V ( G ) = { w , w , w } . Please note that the sets { w , w , w } and { u , u , u , v , v , v } may intersect.By Lemma 4.2 applied to the path v P w there exist two disjoint paths Q , Q in G from Y to V ( P ∪ P ) ∪ { w } , with no internal vertices in Y ∪ V ( P ∪ P ) and such that one of them, say Q , endsin w . From the symmetry we may assume that Q ends in V ( P ). Similarly as before, let C denotethe subpath of C \ u with ends u and u , and let C and C be defined similarly. If C has at least l/ P ∪ Q and P ∪ Q ∪ C ∪ w P u include a linkage from X to Y of length at least l/
5, as desired. Thus we may assume that | E ( C ) | < l/
5. If u V ( Q ), thenreplacing C by C ∪ C above results in a linkage from X to Y of length at least 4 l/
5. Thus we mayassume that V ( Q ) ∩ V ( P ) = { w } and w = u . But now either ( P ∪ Q , P ∪ C ∪ Q , C , C )is a hammock from X to Y of length at least 2 l/
5, or ( P ∪ Q , P ∪ C ∪ Q , C , C ) is a hammockfrom Y to X of length at least 2 l/
5, as desired.
Lemma 4.4
Let G be a -connected graph, let X, Y ⊆ V ( G ) be disjoint sets of size two, and let G have a hammock from X to Y of length l . Then G has either a non-singular hammock from X to Y r from Y to X of length at least l/ , or a linkage from X to Y of length at least l/ . Proof:
Let η = ( P , P , R , R ) be a hammock in G of length l , and let x , x , y , y , s , s , t , t be as in the definition of hammock. We may assume that s = s , for otherwise η is non-singular,and hence satisfies the conclusion of the lemma. Since x = y , at least one of the sets A := V ( x P s ) − { s } and B := V ( y P s ) − { s } is not empty.Assume first that A = ∅ . Since G is 3-connected, there is a path Q in G \{ s , t } with ends a ∈ A and b ∈ V ( P ∪ R ∪ R ∪ s P y ). If b ∈ V ( R ∪ t P y ), then x P s ∪ Q ∪ R ∪ t P y includes apath from x to y that together with x P t ∪ R ∪ s P y forms a linkage from X to Y of length atleast l . If b ∈ V ( x P t ), then ( P , P , R , Q ) is a non-singular hammock from Y to X of length l . If b ∈ V ( s P y ), then the paths x P a ∪ Q ∪ bP y and x P t ∪ R ∪ R ∪ t P y form a linkage from X to Y of length at least l . Thus we may assume that b ∈ V ( R ). If the path t R b has at least l/ P , P , Q ∪ t R b, R ) is a non-singular hammock from X to Y of length at least l/ s R b has at least l/ P and x P a ∪ Q ∪ bR s ∪ s P y forma linkage from X to Y of length at least l/
2. This completes the case A = ∅ .Thus we may assume that B = ∅ . We take a path in G \ { s , t } connecting a vertex in B to avertex in V ( P ∪ R ∪ R ∪ x P s ) and proceed similarly as in the previous paragraph. The detailsare analogous to the case A = ∅ and are left to the reader. Lemma 4.5
Let G be a -connected graph, let X, Y ⊆ V ( G ) be disjoint sets of size two, and assumethat G has a non-singular hammock from X to Y of length l . Then G has a linkage from X to Y oflength at least l/ . Proof:
Let η = ( P , P , R , R ) be a non-singular hammock in G , and let x , x , y , y , s , s , t , t be as in the definition of hammock. Since G is non-singular, ( P , P , R , R ) is also a hammockfrom X to Y of the same length as η , and hence there is symmetry between P and P . Let A := x P s ∪ R ∪ x P t and B := y P s ∪ R ∪ y P t . Then A and B are paths in G . ByLemma 4.2 applied to the sets V ( A ) and V ( B ) and paths s P s and t P t there exist three disjointpaths Q, Q , Q from V ( A ) to V ( B ) such that two of them have ends in { s , t } , and two have endsin { s , t } . From the symmetry we may assume that s is an end of Q ; let s ′ be the other end of Q . Similarly we may assume that t is an end of Q ; let t ′ be the other end of Q . Now the path x P s ∪ Q ∪ s ′ By can play the role of P , the path x P t ∪ Q ∪ t ′ By can play the role of P , andthe path s ′ Bt ′ can play the role of R . In other words, we may assume (by changing the hammock η but not changing its length) that there exists a path Q from V ( A ) to V ( B ) that is disjoint fromthe paths P and P . Let a be the end of Q in V ( A ), and let b be the end of Q in V ( B ). From thesymmetry we may assume that either a ∈ V ( x P s ), or a ∈ V ( R ) and the path aR t has at least l/ a ∈ V ( x P s ). If b ∈ V ( s P y ), then the paths x P a ∪ Q ∪ bP y and x P t ∪ R ∪ s P s ∪ R ∪ t P y form a linkage from X to Y of length at least l , and if b ∈ V ( R ∪ t P y ), then the path x P t ∪ R ∪ s P y and a subpath of x P a ∪ Q ∪ R ∪ t P y forma linkage from X to Y of length at least l . This completes the case when a ∈ V ( x P s ).We may therefore assume that a ∈ V ( R ) and the path aR t has at least l/ b ∈ V ( s P y ), then the paths x P s ∪ R ∪ t P y and x P t ∪ t R a ∪ Q ∪ bP y form a linkagefrom X to Y of length at least l/
2, and if b ∈ V ( R ∪ t P y ), then the path P and a subpath of x P t ∪ t R a ∪ Q ∪ R ∪ t P y form a linkage from X to Y of length at least l/ Proof of Lemma 4.1.
Let
G, X, Y be as stated, and assume that G has a path of length l . Then G has a cycle of length at least 2 l/ G hasa hammock from X to Y of length at least 4 l/
25, for otherwise the theorem holds. Similarly, byLemma 4.4 we may assume that G has a non-singular hammock from X to Y of length at last 2 l/ G has a linkage from X to Y of length at least l/
25, as desired.
Notation.
Throughout this section we will assume the following notation. Let k ≥ G be a k -critical graph on n vertices, and let ( T, W ) be a standard tree-decomposition of G . Oneexists by Lemma 3.1. For t ∈ V ( T ) let H t denote the torso of ( G, T, W ) at t , and let N t denote thenucleus of ( G, T, W ) at t . We select a vertex r ∈ V ( T ) of degree one that we will regard as the rootof T . Thus a descendant of a vertex t ∈ V ( T ) is any vertex t ′ ∈ V ( T ) − { t } such that t belongs tothe path from r to t ′ in T . For t ∈ V ( T ) we denote by T t the subtree of T induced by t and all itsdescendants. We define a weight function w : V ( T ) → { , , . . . } by w ( t ) := | E ( N t ) | . Thus w ( t ) ≥ t ∈ V ( T ) by Lemma 3.4(ii). According to the convention introduced prior to lemma 3.10, w ( T t ) means P v ∈ V ( T t ) w ( t ). We now define, for every t ∈ V ( T ), a set X t ⊆ W t of size two. If t = r ,then let t ′ be the parent of t in the rooted tree ( T, r ) and we set X t = W t ∩ W t ′ . If t = r and T hasat least two vertices, then let t ′ be the unique child of r in T and let X r ⊆ W r be any set disjointfrom W t ∩ W t ′ that consists of two vertices that are adjacent in G . Such a set exists because H t is3-connected by Lemma 3.6 and the elements of W t ∩ W t ′ form the only edge of H t that does notbelong to G . Finally, if T has only one vertex we choose X t arbitrarily. For t ∈ V ( T ) we denote by G t the graph induced in G by the set of vertices S t ′ ∈ V ( T t ) W t ′ .In order to be able to apply Lemma 3.10 we prove the following lemma. Lemma 5.1
Let t ∈ V ( T ) − { r } , and let X t = { x, x ′ } . Then the graph G t \ x has a spanning tree R such that for every integer l ≥ there are at most k l vertices of R at distance exactly l from x ′ . roof. Let G ′ t be the subgraph of G induced by the union of all W t ′ over all t ′ ∈ V ( T ) − V ( T t ).Then G t ∩ G ′ t = X t = W t ∩ W t ′ , where t ′ is the ancestor of t in the rooted tree ( T, r ). By Lemma3.2(iv) applied to G and the vertices x, x ′ the graph G t was obtained from some k -critical graph H by either (i) deleting the edge xx ′ , or (ii) splitting a vertex of H into the two vertices x, x ′ .Assume first that G t was obtained by deleting the edge xx ′ . Since H is 2-connected by Lemma 3.2(i),the graph H \ x has a DFS spanning tree R rooted at x ′ . We deduce from Lemma 2.1 applied to H and X = { x } that R satisfies the conclusion of the lemma.We may therefore assume that G t was obtained from H by splitting a vertex, say z , into the twovertices x, x ′ . Since G is 2-connected by Lemma 3.2(i), there is an edge e ∈ E ( G t ) joining the vertex x ′ to a vertex in V ( G t ) − { x, x ′ } . Then e is also an edge of H . Since H is 2-connected, it has a DFSspanning tree R ′ rooted at z such that e is the only edge of R ′ incident with z . The tree R ′ givesrise to a unique spanning tree R of G t \ x with the same edge-set in the obvious way. It follows fromLemma 2.1 applied to H and X = ∅ that R satisfies the conclusion of the lemma.To prove Theorem 1.1 we prove, for the sake of induction, the following lemma. Lemma 5.2
For every t ∈ V ( T ) the graph G t has a path connecting the vertices of X t of length atleast log w ( T t ) / (100 log k ) . Let us first derive Theorem 1.1 from Lemma 5.2.
Proof of Theorem 1.1, assuming Lemma 5.2.
Apply Lemma 5.2 with t = r . Since by definitionthe vertices of X r are adjacent, we get a cycle of length at least log w ( T ) / (100 log k ). Since distinctnuclei are edge-disjoint by the definition of nucleus, Lemma 3.8 implies w ( T ) = P t ∈ V ( T ) | E ( N t ) | ≥| E ( G ) | ≥ n , and hence G has a cycle of length at least log n/ (100 log k ).The rest of this section is devoted to a proof of Lemma 5.2. We first take care of the followingspecial case. Lemma 5.3
Let t ∈ V ( T ) . The statement of Lemma 5.2 holds for t if w ( t ) ≥ w ( T t ) / . In particular,the lemma holds for t if | V ( T t ) | = 1 . Proof.
The second assertion follows from the first, and so it suffices to prove the first statement. ByLemma 3.9 the torso H t has a path of length at least log w ( t ) / log k (recall that w ( t ) is the numberof edges in the nucleus N t ). Lemma 3.6 guarantees that H t is 3-connected, and so by Theorem 2.4we get that H t has a cycle C of length at least log w ( t ) / log k . Since H t is 3-connected we get fromMenger’s Theorem that it contains two disjoint paths connecting X t to C . Suppose these paths meet C at vertices w, w ′ . Then one of the two subpaths of C connecting w and w ′ has length at least log w ( t ) / log k . Together with the two paths connecting the vertices of X t to C we get a path P H t connecting the vertices of X t of length at least log w ( t ) / log k ≥ log w ( T t ) / log k by thehypothesis of the lemma and the fact that w ( T t ) ≥ e = uv ∈ E ( P ) − E ( G ) we do the following. By Lemma 3.3(ii) there is a uniqueneighbor t ′ of t in T such that W t ∩ W t ′ = { u, v } . If r = t , then t ′ is not the parent of t in the rootedtree ( T, r ), because { u, v } 6 = X t by the choice of P . We claim that there exists a path P e in G t ′ with ends u, v . Indeed, since ( T, W ) is standard, there exists a vertex w ∈ W t ′ − { u, v } . Since G is2-connected by Lemma 3.2(i), there exist two paths P , P in G with one end w and the other end in { u, v } , pairwise disjoint, except for w . Then P e := P ∪ P is a path in G with ends u, v . It follows that P e is a path in G t ′ , for otherwise some subpath Q of P e \{ u, v } joins the vertex w ∈ V ( G t ′ ) − { u, v } to a vertex of V ( G ) − V ( G t ′ ). But then Q has an edge with one end in V ( G t ′ ) − { u, v } and the otherend in V ( G ) − V ( G t ′ ), contrary to definition of tree-decomposition. This proves our claim that P e exists. We replace e by the path P e and repeat the construction for each edge e ∈ E ( P ) − E ( G ). Fordistinct edges e, e ′ ∈ E ( P ) − E ( G ) the paths P e , P e ′ have no internal vertices in common, becausetheir interiors belong to disjoint subgraphs. We thus arrive at a path in G with ends in X t of lengthat least log n/ (100 log k ), as desired. Proof of Lemma 5.2.
We proceed by induction on | V ( T t ) | . Since Lemma 5.3 establishes the basecase | V ( T t ) | = 1, we can assume henceforth that | V ( T t ) | > | V ( T t ) | . Let X t = { x, x ′ } , let N be the children of t in the rooted tree ( T, r ) and define N = { t ′ ∈ N : W t ∩ W t ′ ∩ X t = ∅} N = { t ′ ∈ N : W t ∩ W t ′ ∩ X t = { x }} N = { t ′ ∈ N : W t ∩ W t ′ ∩ X t = { x ′ }} The sets N , N , N form a partition of N . For t = r this follows from Lemma 3.3(ii), and for t = r this follows from the way we picked X r . Therefore, either X y ∈ N w ( T y ) ≥
34 ( w ( T t ) − w ( t )) , (10)or X y ∈ N ∪ N w ( T y ) ≥
14 ( w ( T t ) − w ( t )) . (11)We first deal with the case (10). By Lemma 5.3 we may assume that w ( t ) < w ( T t ) /
5, and hence X y ∈ N w ( T y ) ≥ w ( T t ) . (12)16y Lemma 3.7 we know that | N | ≤ | N | ≤ w ( t ). Therefore, there is a vertex t ′ ∈ N for which w ( T t ′ ) ≥ w ( T t )5 w ( t ) . (13)By Lemma 3.9 the graph H t (the torso at t ) has a path of length at least log w ( t ) / log k . Therefore,by Lemma 3.6, we can apply Lemma 4.1 to the graph H t and sets X t and X t ′ to deduce that H t hastwo disjoint paths P , P from X t to X t ′ satisfying | E ( P ) | + | E ( P ) | ≥ log w ( t )50 log k . (14)By the induction hypothesis the graph G t ′ has a path P connecting the pair of vertices of X t ′ satisfying | E ( P ) | ≥ log w ( T t ′ )100 log k . (15)Combining (13), (14) and (15) we get that P ∪ P ∪ P is a path in G t ∪ H t with ends in X t of lengthat least | E ( P ) | + | E ( P ) | + | E ( P ) | ≥ log w ( t )50 log k + log w ( T t ′ )100 log k ≥ log w ( t )50 log k + log w ( T t )100 log k − log w ( t ) + log 5100 log k ≥ log w ( T t )100 log k , because w ( t ) ≥
6. We now convert P ∪ P ∪ P to a path in G t of length at least log w ( T t ) / (100 log k )in the same way as in the second paragraph of the proof of Lemma 5.3. This completes the proofwhen (10) holds.Thus we may assume (11). From the symmetry between N and N we may assume that X y ∈ N w ( T y ) ≥ ( w ( T t ) − w ( t )) / . Again, by Lemma 5.3 we may assume that w ( t ) < w ( T t ) /
5, and hence X y ∈ N w ( T y ) ≥ w ( T t ) / . (16)It follows that t = r , for otherwise N = ∅ . We need to define a new weight function φ : V ( G t ) − { x } → { , , . . . } . Let v ∈ V ( G t ) − { x } . If v ∈ W t and there exists a neighbor t ′ of t in T t such that W t ′ ∩ W t = { x, v } , then t ′ is unique by Lemma 3.3(ii), and we define φ ( v ) = w ( T t ′ ). If v W t or no such t ′ exists, then we define φ ( v ) = 0. Thus, in particular, φ ( x ′ ) = 0 by Lemma 3.3(ii).By Lemma 5.1 the graph G t \ x has a spanning tree R such that for every integer l ≥ k l vertices of R at distance exactly l from x ′ . Note that by (16) we have that the total weightof R satisfies φ ( R ) = X y ∈ N w ( T y ) ≥ w ( T t ) / . (17)By Lemma 3.10 applied to the tree R and vertex x ′ there exists a vertex v ∈ V ( R ) at distance l from x ′ in R such that φ ( v ) > l log k + log φ ( v ) ≥ log φ ( T ). It follows that there is a path P in G t \ x from x ′ to v satisfying 2 | E ( P ) | log k + log φ ( v ) ≥ log φ ( R ) . (18)Since φ ( v ) > v ∈ W t and P has length at least one. Let t ′ ∈ N be such that W t ′ ∩ W t = { x, v } , so that φ ( v ) = w ( T t ′ ). Since P is a path from x ′ ∈ W t − V ( G t ′ ) to v in G t \ x , wededuce that V ( P ) ∩ V ( G t ′ ) = { v } . By the induction hypothesis applied to the graph G t ′ the graph G t ′ has a path Q connecting x to v of length at least log φ ( v ) / log k . So P ∪ Q is a path in G t from x to x ′ of length at least | E ( P ) | + log φ ( v )100 log k = 150 (cid:18) | E ( P ) | + log φ ( v )2 log k (cid:19) + (cid:18) − (cid:19) | E ( P ) |≥ log φ ( R )100 log k + 1 − ≥ log( w ( T t ) / k + 1 − w ( T t )100 log k − log 10100 log k + 1 − ≥ log w ( T t )100 log k , where in the first inequality we used (18) and the fact that | E ( P ) | ≥
1, and the second inequalityuses (17). This completes the proof of Lemma 5.2.
We need to introduce the notion of Haj´os sum of two graphs. Let K and L be two graphs withdisjoint vertex-sets, and let k k and l l be edges of K and L , respectively. Let G be the graphobtained from the union of K and L by deleting the edges k k and l l , identifying the vertices k and l , and adding an edge joining k and l . In those circumstances we say that G is a Haj´os sum of K and L . It is straightforward to check that if K and L are k -critical, then so is G .We now describe a construction of k -critical graphs with no long path, and hence no long cycle.Let k ≥ T be a tree of maximum degree at most k −
1, and let ( H t : t ∈ V ( T ))be a family of k -critical graphs, each containing the same vertex x , and otherwise pairwise disjoint.For every ordered pair t, t ′ of adjacent vertices of T we select a vertex v tt ′ ∈ V ( H t ) such that18 v tt ′ is adjacent to x in H t , and • if t ′ and t ′′ are distinct neighbors of t in T , then v tt ′ = v tt ′′ .Such a choice is possible, because T has maximum degree at most k − k -critical graphhas minimum degree at least k −
1. Let us emphasize that even though tt ′ and t ′ t denote the sameedge of T , the vertices v tt ′ and v t ′ t are distinct: the first belongs to H t and the second to H t ′ . Wedefine a graph G to be the graph obtained from S t ∈ V ( T ) H t by, for every edge tt ′ ∈ E ( T ), deletingthe edges x v tt ′ and x v t ′ t , and adding an edge joining v tt ′ and v t ′ t .It is easy to see that the graph G can be viewed as being obtained from the graphs H t byrepeatedly taking Haj´os sums, and is thus k -critical. Also, it has 1 + P t ∈ V ( T ) ( | V ( H t ) | −
1) vertices,and for every path P in G \ x there exists a path R in T such that | V ( P ) | ≤ P t ∈ V ( R ) ( | V ( H t ) | − h ≥ T be the ( k − h ; that is, a tree T with a vertex r ∈ V ( T ) such that everyvertex is at distance at most h from r , and each vertex at distance at most h − r has degreeexactly k −
1. Each of the graphs H t will be the complete graph on k vertices. Then the graphsresulting from the construction described above with this choice of T and H t prove the inequality (5)and statement (6), as is easily seen. (In Gallai’s original construction the vertex r has degree k − k -critical graphs on n vertices for every n ≥ k ,except n = k + 1. It is easy to deduce the following theorem, by utilizing such k -critical graphs andtrees that are not necessarily regular, and the above construction. Theorem 6.1
For every integer k ≥ and every integer n ≥ k + 2 L k ( n ) ≤ k − k −
1) log n + 2 k . (19) Acknowledgment.
We are grateful to Michael Krivelevich for answering many questions relatedto this paper and especially for sharing his English translation of the construction of Gallai from [7].
References [1] N. Alon, M. Krivelevich and P. D. Seymour, Long cycles in critical graphs, J. Graph Theory 35(2000), 193-196.[2] A. Bondy and S. C. Locke, Relative lengths of paths and cycles in 3-connected graphs, DiscreteMath. 33 (1981), 111-122.[3] R. Diestel, Graph theory, third edition, Springer, Berlin, 2005.194] G. A. Dirac, On the structure of 5- and 6-chromatic abstract graphs, J. Reine Angew. Math.214/215 (1964) 43-52.[5] G. A. Dirac, Some theorems on abstract graphs, Proc. of the London Math. Soc., 2 (1952),69-81.[6] G. A. Dirac, Circuits in critical graphs, Monatsh. Math. 59 (1955) 178-187.[7] T. Gallai, Kritische Graphen I, Publ. Math. Inst. Hungar. Acad. Sci. 8 (1963), 165-192.[8] T. Jensen and B. Toft,
Graph Coloring Problems , Wiley, New York, 1995.[9] J. B. Kelly and L. M. Kelly, Paths and circuits in critical graphs, Amer. J. Math. 76 (1954),786-792.[10] J. Bang-Jensen, B. Reed, M. Schacht, R. ˇS´amal, B. Toft and U. Wagner, On six problemsposed by Jarik Neˇsetˇril, In: Topics in Discrete Mathematics. Dedicated to Jarik Neˇsetˇril on theoccasion of his 60th birthday. M. Klazar, J. Kratochv´ıl, M. Loebl, J. Matouˇsek, R. Thomas andP. Valtr, editors,
Algorithms and Combinatorics26