A Dirac type result on Hamilton cycles in oriented graphs
aa r X i v : . [ m a t h . C O ] J un A DIRAC TYPE RESULT ON HAMILTON CYCLES IN ORIENTEDGRAPHS
LUKE KELLY, DANIELA K ¨UHN AND DERYK OSTHUS
Abstract.
We show that for each α > G with δ + ( G ) , δ − ( G ) ≥ | G | / α | G | contains a Hamilton cycle. This gives an approximatesolution to a problem of Thomassen [21]. In fact, we prove the stronger result that G isstill Hamiltonian if δ ( G ) + δ + ( G ) + δ − ( G ) ≥ | G | / α | G | . Up to the term α | G | thisconfirms a conjecture of H¨aggkvist [10]. We also prove an Ore-type theorem for orientedgraphs. Introduction An oriented graph G is obtained from a (simple) graph by orienting its edges. Thusbetween every pair of vertices of G there exists at most one edge. The minimum semi-degree δ ( G ) of G is the minimum of its minimum outdegree δ + ( G ) and its minimum in-degree δ − ( G ). When referring to paths and cycles in oriented graphs we always mean thatthese are directed without mentioning this explicitly.A fundamental result of Dirac states that a minimum degree of | G | / G . There is an analogue of this for digraphs dueto Ghouila-Houri [9] which states that every digraph D with minimum semi-degree atleast | D | / G . This question was first raisedby Thomassen [20], who [22] showed that a minimum semi-degree of | G | / − p | G | / G is not far from beinga tournament. H¨aggkvist [10] improved the bound further to | G | / − − | G | and conjec-tured that the actual value lies close to 3 | G | /
8. The best previously known bound is dueto H¨aggkvist and Thomason [11], who showed that for each α > G with minimum semi-degree at least (5 /
12 + α ) | G | has a Hamilton cycle.Our first result implies that the actual value is indeed close to 3 | G | / Theorem 1.
For every α > there exists an integer N = N ( α ) such that every orientedgraph G of order | G | ≥ N with δ ( G ) ≥ (3 / α ) | G | contains a Hamilton cycle. A construction of H¨aggkvist [10] shows that the bound in Theorem 1 is essentially bestpossible (see Proposition 6).In fact, H¨aggkvist [10] formulated the following stronger conjecture. Given an orientedgraph G , let δ ( G ) denote the minimum degree of G (i.e. the minimum number of edgesincident to a vertex) and set δ ∗ ( G ) := δ ( G ) + δ + ( G ) + δ − ( G ). Conjecture 2 (H¨aggkvist [10]) . Every oriented graph G with δ ∗ ( G ) > (3 n − / has aHamilton cycle. D. K¨uhn was partially supported by the EPSRC, grant no. EP/F008406/1. D. Osthus was partiallysupported by the EPSRC, grant no. EP/E02162X/1 and EP/F008406/1.
Our next result provides an approximate confirmation of this conjecture for large orientedgraphs.
Theorem 3.
For every α > there exists an integer N = N ( α ) such that every orientedgraph G of order | G | ≥ N with δ ∗ ( G ) ≥ (3 / α ) | G | contains a Hamilton cycle. Note that Theorem 1 is an immediate consequence of this. The proof of Theorem 3 can bemodified to yield the following Ore-type analogue of Theorem 1. (Ore’s theorem [19] statesthat every graph G on n ≥ d ( x ) + d ( y ) ≥ n whenever xy / ∈ E ( G )has a Hamilton cycle.) Theorem 4.
For every α > there exists an integer N = N ( α ) such that every orientedgraph G of order | G | ≥ N with d + ( x ) + d − ( y ) ≥ (3 / α ) | G | whenever xy / ∈ E ( G ) containsa Hamilton cycle. A version for general digraphs was proved by Woodall [23]: every strongly connecteddigraph D on n ≥ d + ( x ) + d − ( y ) ≥ n whenever xy / ∈ E ( D ) has aHamilton cycle.Theorem 1 immediately implies a partial result towards a conjecture of Kelly (see e.g. [3]),which states that every regular tournament on n vertices can be partitioned into ( n − / Corollary 5.
For every α > there exists an integer N = N ( α ) such that every regulartournament of order n ≥ N contains at least (1 / − α ) n edge-disjoint Hamilton cycles. Indeed, Corollary 5 follows from Theorem 1 by successively removing Hamilton cycles untilthe oriented graph G obtained from the tournament in this way has minimum semi-degreeless than (3 / α ) | G | . The best previously known bound on the number of edge-disjointHamilton cycles in a regular tournament is the one which follows from the result of H¨aggkvistand Thomason [11] mentioned above. A related result of Frieze and Krivelevich [8] statesthat every dense ε -regular digraph contains a collection of edge-disjoint Hamilton cycleswhich covers almost all of its edges. This immediately implies that the same holds foralmost every tournament. Together with a lower bound by McKay [18] on the number ofregular tournaments, the above result in [8] also implies that almost every regular tournamentcontains a collection of edge-disjoint Hamilton cycles which covers almost all of its edges.Note that Theorem 3 implies that for sufficiently large tournaments T a minimum semi-degree of at least (1 / α ) | T | already suffices to guarantee a Hamilton cycle. (However, itis not hard to prove this directly.) It was shown by Bollob´as and H¨aggkvist [5] that thisdegree condition even ensures the k th power of a Hamilton cycle (if T is sufficiently largecompared to 1 /α and k ). The degree condition is essentially best possible as a minimumsemi-degree of | T | / − DIRAC TYPE RESULT ON HAMILTON CYCLES IN ORIENTED GRAPHS 3 we collect some preliminary results. Theorem 3 is then proved in Section 5. In the lastsection we discuss the modifications needed to prove Theorem 4.2.
Notation and the extremal example
Before we show that Theorems 1, 3 and 4 are essentially best possible, we will introducethe basic notation used throughout the paper. Given two vertices x and y of an orientedgraph G , we write xy for the edge directed from x to y . The order | G | of G is the number of itsvertices. We write N + G ( x ) for the outneighbourhood of a vertex x and d + G ( x ) := | N + G ( x ) | forits outdegree. Similarly, we write N − G ( x ) for the inneighbourhood of x and d − G ( x ) := | N − G ( x ) | for its indegree. We write N G ( x ) := N + G ( x ) ∪ N − G ( x ) for the neighbourhood of x anduse N + ( x ) etc. whenever this is unambiguous. We write ∆( G ) for the maximum of | N ( x ) | over all vertices x ∈ G .Given a set A of vertices of G , we write N + G ( A ) for the set of all outneighbours of verticesin A . So N + G ( A ) is the union of N + G ( a ) over all a ∈ A . N − G ( A ) is defined similarly. Theoriented subgraph of G induced by A is denoted by G [ A ]. Given two vertices x, y of G , an x - y path is a directed path which joins x to y . Given two disjoint subsets A and B of verticesof G , an A - B edge is an edge ab where a ∈ A and b ∈ B , the set of these edges is denotedby E G ( A, B ) and we put e G ( A, B ) := | E G ( A, B ) | .Recall that when referring to paths and cycles in oriented graphs we always mean thatthey are directed without mentioning this explicitly. Given two vertices x and y on a directedcycle C , we write xCy for the subpath of C from x to y . Similarly, given two vertices x and y on a directed path P such that x precedes y , we write xP y for the subpath of P from x to y .A walk in an oriented graph G is a sequence of (not necessary distinct) vertices v , v , . . . , v ℓ where v i v i +1 is an edge for all 1 ≤ i < ℓ . The walk is closed if v = v ℓ . A 1 -factor of G is acollection of disjoint cycles which cover all the vertices of G . We define things similarly forgraphs and for directed graphs. The underlying graph of an oriented graph G is the graphobtained from G by ignoring the directions of its edges.Given disjoint vertex sets A and B in a graph G , we write ( A, B ) G for the induced bipartitesubgraph of G whose vertex classes are A and B . We write ( A, B ) where this is unambiguous.We call an orientation of a complete graph a tournament and an orientation of a completebipartite graph a bipartite tournament . An oriented graph G is d-regular if all vertices havein- and outdegree d . G is regular if it is d -regular for some d . It is easy to see (e.g. byinduction) that for every odd n there exists a regular tournament on n vertices. Throughoutthe paper we omit floors and ceilings whenever this does not affect the argument.The following construction by H¨aggkvist [10] shows that Conjecture 2 is best possible forinfinitely many values of | G | . We include it here for completeness. Proposition 6.
There are infinitely many oriented graphs G with minimum semi-degree (3 | G |− / which do not contain a -factor and thus do not contain a Hamilton cycle. Proof.
Let n := 4 m + 3 for some odd m ∈ N . Let G be the oriented graph obtainedfrom the disjoint union of two regular tournaments A and C on m vertices, a set B of m + 2vertices and a set D of m + 1 vertices by adding all edges from A to B , all edges from B to C , all edges from C to D as well as all edges from D to A . Finally, between B and D we add edges to obtain a bipartite tournament which is as regular as possible, i.e. the in-and outdegree of every vertex differ by at most 1. So in particular every vertex in B sendsexactly ( m + 1) / D (Figure 1).It is easy to check that the minimum semi-degree of G is ( m − / m + 1) = (3 n − / B has to pass through D , it follows LUKE KELLY, DANIELA K ¨UHN AND DERYK OSTHUS
PSfrag replacements
AB CD
Figure 1.
The oriented graph in the proof of Proposition 6.that every cycle contains at least as many vertices from D as it contains from B . As | B | > | D | this means that one cannot cover all the vertices of G by disjoint cycles, i.e. G does not contain a 1-factor. (cid:3) The Diregularity lemma, the Blow-up lemma and other tools
The Diregularity lemma and the Blow-up lemma.
In this section we collect allthe information we need about the Diregularity lemma and the Blow-up lemma. See [16] fora survey on the Regularity lemma and [14] for a survey on the Blow-up lemma. We startwith some more notation. The density of a bipartite graph G = ( A, B ) with vertex classes A and B is defined to be d G ( A, B ) := e G ( A, B ) | A | | B | . We often write d ( A, B ) if this is unambiguous. Given ε >
0, we say that G is ε -regular if for all subsets X ⊆ A and Y ⊆ B with | X | > ε | A | and | Y | > ε | B | we have that | d ( X, Y ) − d ( A, B ) | < ε . Given d ∈ [0 ,
1] we say that G is ( ε, d )- super-regular if it is ε -regular and furthermore d G ( a ) ≥ ( d − ε ) | B | for all a ∈ A and d G ( b ) ≥ ( d − ε ) | A | for all b ∈ B . (This is a slight variation of the standard definition of ( ε, d )-super-regularity whereone requires d G ( a ) ≥ d | B | and d G ( b ) ≥ d | A | .)The Diregularity lemma is a version of the Regularity lemma for digraphs due to Alonand Shapira [1]. Its proof is quite similar to the undirected version. We will use the degreeform of the Diregularity lemma which can be easily derived (see e.g. [24]) from the standardversion, in exactly the same manner as the undirected degree form. Lemma 7 (Degree form of the Diregularity lemma) . For every ε ∈ (0 , and every inte-ger M ′ there are integers M and n such that if G is a digraph on n ≥ n vertices and d ∈ [0 , is any real number, then there is a partition of the vertices of G into V , V , . . . , V k ,a spanning subdigraph G ′ of G and a set U of ordered pairs V i V j (where ≤ i, j ≤ k and i = j ) such that the following holds: • M ′ ≤ k ≤ M , • | V | ≤ εn , • | V | = · · · = | V k | =: m , • d + G ′ ( x ) > d + G ( x ) − ( d + ε ) n for all vertices x ∈ G , DIRAC TYPE RESULT ON HAMILTON CYCLES IN ORIENTED GRAPHS 5 • d − G ′ ( x ) > d − G ( x ) − ( d + ε ) n for all vertices x ∈ G , • | U | ≤ εk , • for every ordered pair V i V j / ∈ U with ≤ i, j ≤ k and i = j the bipartite graph ( V i , V j ) G whose vertex classes are V i and V j and whose edge set is the set E G ( V i , V j ) of all the V i - V j edges in G is ε -regular, • G ′ is obtained from G by deleting the following edges of G : all edges with bothendvertices in V i for all i ≥ as well as all edges in E G ( V i , V j ) for all V i V j ∈ U andfor all those V i V j / ∈ U with ≤ i, j ≤ k and i = j for which the density of ( V i , V j ) G is less than d . V , . . . , V k are called clusters , V is called the exceptional set and the vertices in V arecalled exceptional vertices . U is called the set of exceptional pairs of clusters . Note that thelast two conditions of the lemma imply that for all 1 ≤ i, j ≤ k with i = j the bipartitegraph ( V i , V j ) G ′ is ε -regular and has density either 0 or density at least d . In particular, in G ′ all pairs of clusters are ε -regular in both directions (but possibly with different densities).We call the spanning digraph G ′ ⊆ G given by the Diregularity lemma the pure digraph .Given clusters V , . . . , V k and the pure digraph G ′ , the reduced digraph R ′ is the digraphwhose vertices are V , . . . , V k and in which V i V j is an edge if and only if G ′ contains a V i - V j edge. Note that the latter holds if and only if ( V i , V j ) G ′ is ε -regular and has density atleast d . It turns out that R ′ inherits many properties of G , a fact that is crucial in our proof.However, R ′ is not necessarily oriented even if the original digraph G is, but the next lemmashows that by discarding edges with appropriate probabilities one can go over to a reducedoriented graph R ⊆ R ′ which still inherits many of the properties of G . (d) will only be usedin the proof of Theorem 4. Lemma 8.
For every ε ∈ (0 , there exist integers M ′ = M ′ ( ε ) and n = n ( ε ) such that thefollowing holds. Let d ∈ [0 , and let G be an oriented graph of order at least n and let R ′ be the reduced digraph and U the set of exceptional pairs of clusters obtained by applying theDiregularity lemma to G with parameters ε , d and M ′ . Then R ′ has a spanning orientedsubgraph R with (a) δ + ( R ) ≥ ( δ + ( G ) / | G | − (3 ε + d )) | R | , (b) δ − ( R ) ≥ ( δ − ( G ) / | G | − (3 ε + d )) | R | , (c) δ ( R ) ≥ ( δ ( G ) / | G | − (3 ε + 2 d )) | R | , (d) if ε ≤ d ≤ − ε and c ≥ is such that d + ( x ) + d − ( y ) ≥ c | G | whenever xy / ∈ E ( G ) then d + R ( V i ) + d − R ( V j ) ≥ ( c − ε − d ) | R | whenever V i V j / ∈ E ( R ) ∪ U . Proof.
Let us first show that every cluster V i satisfies(1) | N R ′ ( V i ) | / | R ′ | ≥ δ ( G ) / | G | − (3 ε + 2 d ) . To see this, consider any vertex x ∈ V i . As G is an oriented graph, the Diregularity lemmaimplies that | N G ′ ( x ) | ≥ δ ( G ) − d + ε ) | G | . On the other hand, | N G ′ ( x ) | ≤ | N R ′ ( V i ) | m + | V | ≤| N R ′ ( V i ) || G | / | R ′ | + ε | G | . Altogether this proves (1).We first consider the case when(2) δ + ( G ) / | G | ≥ ε + d and δ − ( G ) / | G | ≥ ε + d and c ≥ ε + 2 d. Let R be the spanning oriented subgraph obtained from R ′ by deleting edges randomly asfollows. For every unordered pair V i , V j of clusters we delete the edge V i V j (if it exists) withprobability(3) e G ′ ( V j , V i ) e G ′ ( V i , V j ) + e G ′ ( V j , V i ) . LUKE KELLY, DANIELA K ¨UHN AND DERYK OSTHUS
Otherwise we delete V j V i (if it exists). We interpret (3) as 0 if V i V j , V j V i / ∈ E ( R ′ ). So if R ′ contains at most one of the edges V i V j , V j V i then we do nothing. We do this for all unorderedpairs of clusters independently and let X i be the random variable which counts the numberof outedges of the vertex V i ∈ R . Then E ( X i ) = X j = i e G ′ ( V i , V j ) e G ′ ( V i , V j ) + e G ′ ( V j , V i ) ≥ X j = i e G ′ ( V i , V j ) | V i | | V j |≥ | R ′ || G | | V i | X x ∈ V i ( d + G ′ ( x ) − | V | )(4) ≥ ( δ + ( G ′ ) / | G | − ε ) | R | ≥ ( δ + ( G ) / | G | − (2 ε + d )) | R | ( ) ≥ ε | R | . A Chernoff-type bound (see e.g. [2, Cor. A.14]) now implies that there exists a constant β = β ( ε ) such that P ( X i < ( δ + ( G ) / | G | − (3 ε + d )) | R | ) ≤ P ( | X i − E ( X i ) | > ε E ( X i )) ≤ e − β E ( X i ) ≤ e − βε | R | . Writing Y i for the random variable which counts the number of inedges of the vertex V i in R ,it follows similarly that P ( Y i < ( δ − ( G ) / | G | − (3 ε + d )) | R | ) ≤ e − βε | R | . Suppose that c is as in (d). Consider any pair V i V j / ∈ U of clusters such that either V i V j / ∈ E ( R ′ ) or V i V j , V j V i ∈ E ( R ′ ). (Note that each V i V j / ∈ E ( R ) ∪ U satisfies one of theseproperties.) As before, let X i be the random variable which counts the number of outedgesof V i in R and let Y j be the number of inedges of V j in R . Similary as in (4) one can showthat(5) E ( X i + Y j ) ≥ | R ′ || G | | V i | X x ∈ V i ( d + G ′ ( x ) − | V | ) + X y ∈ V j ( d − G ′ ( y ) − | V | ) . To estimate this, we will first show that there is a set M of at least (1 − ε ) | V i | disjoint pairs( x, y ) with x ∈ V i , y ∈ V j and such that xy / ∈ E ( G ). Suppose first that V i V j , V j V i ∈ E ( R ′ ).But then ( V j , V i ) G is ε -regular of density at least d and thus it contains a matching of sizeat least (1 − ε ) | V i | . As G is oriented this matching corresponds to a set M as required. If V i V j / ∈ E ( R ′ ) then ( V i , V j ) G is ε -regular of density less than d (since V i V j / ∈ U ). Thus thecomplement of ( V i , V j ) G is ε -regular of density at least 1 − d and so contains a matching ofsize at least (1 − ε ) | V i | which again corresponds to a set M as required. Together with (5)this implies that E ( X i + Y j ) ≥ | R ′ || G | | V i | X ( x,y ) ∈ M ( d + G ′ ( x ) + d − G ′ ( y ) − | V | ) ≥ | R ′ || G | | V i | ( c − ε + d ) − ε ) | G | (1 − ε ) | V i | ≥ ( c − (5 ε + 2 d )) | R | ( ) ≥ ε | R | . Similarly as before a Chernoff-type bound implies that P ( X i + Y j < ( c − (6 ε + 2 d )) | R | ) ≤ e − βε | R | . As 2 | R | e − βε | R | < M ′ is chosen to be sufficiently large compared to ε , this implies thatthere is some outcome R which satisfies (a), (b) and (d). But N R ′ ( V i ) = N R ( V i ) for every DIRAC TYPE RESULT ON HAMILTON CYCLES IN ORIENTED GRAPHS 7 cluster V i and so (1) implies that δ ( R ) ≥ ( δ ( G ) / | G | − (3 ε + 2 d )) | R | . Altogether this showsthat R is as required in the lemma.If neither of the conditions in (2) hold, then (a), (b) and (d) are trivial and one can obtainan oriented graph R which satisfies (c) from R ′ by arbitrarily deleting one edge from eachdouble edge. If for example only the first of the conditions in (2) holds, then (b) and (d)are trivial. To obtain an oriented graph R which satisfies (a) we consider the X i as before,but ignore the Y i and the sums X i + Y j . Again, N R ′ ( V i ) = N R ( V i ) for every cluster V i andso (c) is also satisfied. The other cases are similar. (cid:3) The oriented graph R given by Lemma 8 is called the reduced oriented graph . The span-ning oriented subgraph G ∗ of the pure digraph G ′ obtained by deleting all the V i - V j edgeswhenever V i V j ∈ E ( R ′ ) \ E ( R ) is called the pure oriented graph . Given an oriented subgraph S ⊆ R , the oriented subgraph of G ∗ corresponding to S is the oriented subgraph obtainedfrom G ∗ by deleting all those vertices that lie in clusters not belonging to S as well asdeleting all the V i - V j edges for all pairs V i , V j with V i V j / ∈ E ( S ).In our proof of Theorem 3 we will also need the Blow-up lemma. Roughly speaking, itstates the following. Let F be a graph on r vertices, let K be a graph obtained from F byreplacing each vertex of F with a cluster and replacing each edge with a complete bipartitegraph between the corresponding clusters. Define G similarly except that the edges of F nowcorrespond to dense ε -super-regular pairs. Then every subgraph H of K which has boundedmaximum degree is also a subgraph in G . In the original version of Koml´os, S´ark¨ozy andSzemer´edi [15] ε has to be sufficiently small compared to 1 /r (and so in particular we cannottake r = | R | ). We will use a stronger (and more technical) version due to Csaba [6], whichallows us to take r = | R | and does not demand super-regularity. The case when ∆ = 3 ofthis is implicit in [7].In the statement of Lemma 9 and later on we write 0 < a ≪ a ≪ a to mean that wecan choose the constants a , a , a from right to left. More precisely, there are increasingfunctions f and g such that, given a , whenever we choose some a ≤ f ( a ) and a ≤ g ( a ),all calculations needed in the proof of Lemma 9 are valid. Hierarchies with more constantsare defined in the obvious way. Lemma 9 (Blow-up Lemma, Csaba [6]) . For all integers ∆ , K , K , K and every positiveconstant c there exists an integer N such that whenever ε, ε ′ , δ ′ , d are positive constants with < ε ≪ ε ′ ≪ δ ′ ≪ d ≪ / ∆ , /K , /K , /K , c the following holds. Suppose that G ∗ is a graph of order n ≥ N and V , . . . , V k is a partitionof V ( G ∗ ) such that the bipartite graph ( V i , V j ) G ∗ is ε -regular with density either or d forall ≤ i < j ≤ k . Let H be a graph on n vertices with ∆( H ) ≤ ∆ and let L ∪ L ∪ · · · ∪ L k be a partition of V ( H ) with | L i | = | V i | =: m for every i = 1 , . . . , k . Furthermore, supposethat there exists a bijection φ : L → V and a set I ⊆ V ( H ) of vertices at distance at least from each other such that the following conditions hold: (C1) | L | = | V | ≤ K dn . (C2) L ⊆ I . (C3) L i is independent for every i = 1 , . . . , k . (C4) | N H ( L ) ∩ L i | ≤ K dm for every i = 1 , . . . , k . (C5) For each i = 1 , . . . , k there exists D i ⊆ I ∩ L i with | D i | = δ ′ m and such that for D := S ki =1 D i and all ≤ i < j ≤ k || N H ( D ) ∩ L i | − | N H ( D ) ∩ L j || < εm. LUKE KELLY, DANIELA K ¨UHN AND DERYK OSTHUS (C6) If xy ∈ E ( H ) and x ∈ L i , y ∈ L j where i, j = 0 then ( V i , V j ) G ∗ is ε -regular withdensity d . (C7) If xy ∈ E ( H ) and x ∈ L , y ∈ L j then | N G ∗ ( φ ( x )) ∩ V j | ≥ cm . (C8) For each i = 1 , . . . , k , given any E i ⊆ V i with | E i | ≤ ε ′ m there exists a set F i ⊆ ( L i ∩ ( I \ D )) and a bijection φ i : E i → F i such that | N G ∗ ( v ) ∩ V j | ≥ ( d − ε ) m whenever N H ( φ i ( v )) ∩ L j = ∅ (for all v ∈ E i and all j = 1 , . . . , k ). (C9) Writing F := S ki =1 F i we have that | N H ( F ) ∩ L i | ≤ K ε ′ m .Then G ∗ contains a copy of H such that the image of L i is V i for all i = 1 , . . . , k and theimage of each x ∈ L is φ ( x ) ∈ V . The additional properties of the copy of H in G ∗ are not included in the statement of thelemma in [6] but are stated explicitly in the proof.Let us briefly motivate the conditions of the Blow-up lemma. The embedding of H into G guaranteed by the Blow-up lemma is found by a randomized algorithm which first embedseach vertex x ∈ L to φ ( x ) and then successively embeds the remaining vertices of H . So theimage of L will be the exceptional set V . Condition (C1) requires that there are not toomany exceptional vertices and (C2) ensures that we can embed the vertices in L withoutaffecting the neighbourhood of other such vertices. As L i will be embedded into V i we needto have (C3). Condition (C5) gives us a reasonably large set D of ‘buffer vertices’ which willbe embedded last by the randomized algorithm. (C6) requires that edges between verticesof H − L are embedded into ε -regular pairs of density d . (C7) ensures that the exceptionalvertices have large degree in all ‘neighbouring clusters’. (C8) and (C9) allow us to embedthose vertices whose set of candidate images in G ∗ has grown very small at some point ofthe algorithm. Conditions (C6), (C8) and (C9) correspond to a substantial weakening of thesuper-regularity that the usual form of the Blow-up lemma requires, namely that whenever H contains an edge xy with and x ∈ L i , y ∈ L j then ( V i , V j ) G ∗ is ( ε, d )-super-regular.We would like to apply the Blow-up lemma with G ∗ being obtained from the underlyinggraph of the pure oriented graph by adding the exceptional vertices. It will turn out thatin order to satisfy (C8), it suffices to ensure that all the edges of a suitable 1-factor inthe reduced oriented graph R correspond to ( ε, d )-superregular pairs of clusters. A well-known simple fact (see the first part of the proof of Proposition 10) states that this can beensured by removing a small proportion of vertices from each cluster V i , and so (C8) willbe satisfied. However, (C6) requires all the edges of R to correspond to ε -regular pairs ofdensity precisely d and not just at least d . (As remarked by Csaba [6], it actually sufficesthat the densities are close to d in terms of ε .) The second part of the following propositionshows that this too does not pose a problem. Proposition 10.
Let M ′ , n , D be integers and let ε, d be positive constants such that /n ≪ /M ′ ≪ ε ≪ d ≪ /D . Let G be an oriented graph of order at least n . Let R be the reducedoriented graph and let G ∗ be the pure oriented graph obtained by successively applying firstthe Diregularity lemma with parameters ε , d and M ′ to G and then Lemma 8. Let S be anoriented subgraph of R with ∆( S ) ≤ D . Let G ′ be the underlying graph of G ∗ . Then onecan delete Dε | V i | vertices from each cluster V i to obtain subclusters V ′ i ⊆ V i in such a waythat G ′ contains a subgraph G ′ S whose vertex set is the union of all the V ′ i and such that • ( V ′ i , V ′ j ) G ′ S is ( √ ε, d − Dε ) -superregular whenever V i V j ∈ E ( S ) , • ( V ′ i , V ′ j ) G ′ S is √ ε -regular and has density d − Dε whenever V i V j ∈ E ( R ) . Proof.
Consider any cluster V i ∈ V ( S ) and any neighbour V j of V i in S . Recall that m = | V i | . Let d ij denote the density of the bipartite subgraph ( V i , V j ) G ′ of G ′ induced by V i and V j . So d ij ≥ d and this bipartite graph is ε -regular. Thus there are at most 2 εm vertices DIRAC TYPE RESULT ON HAMILTON CYCLES IN ORIENTED GRAPHS 9 v ∈ V i such that || N G ′ ( v ) ∩ V j | − d ij m | > εm . So in total there are at most 2 Dεm vertices v ∈ V i such that || N G ′ ( v ) ∩ V j | − d ij m | > εm for some neighbour V j of V i in S . Delete allthese vertices as well as some more vertices if necessary to obtain a subcluster V ′ i ⊆ V i of size(1 − Dε ) m =: m ′ . Delete any 2 Dεm vertices from each cluster V i ∈ V ( R ) \ V ( S ) to obtaina subcluster V ′ i . It is easy to check that for each edge V i V j ∈ E ( R ) the graph ( V ′ i , V ′ j ) G ′ isstill 2 ε -regular and that its density d ′ ij satisfies d ′ := d − Dε < d ij − ε ≤ d ′ ij ≤ d ij + ε. Moreover, whenever V i V j ∈ E ( S ) and v ∈ V ′ i we have that( d ij − Dε ) m ′ ≤ | N G ′ ( v ) ∩ V ′ j | ≤ ( d ij + 4 Dε ) m ′ . For every pair V i , V j of clusters with V i V j ∈ E ( S ) we now consider a spanning randomsubgraph G ′ ij of ( V ′ i , V ′ j ) G ′ which is obtained by choosing each edge of ( V ′ i , V ′ j ) G ′ with prob-ability d ′ /d ′ ij , independently of the other edges. Consider any vertex v ∈ V ′ i . Then theexpected number of neighbours of v in V ′ j (in the graph G ′ ij ) is at least ( d ij − Dε ) d ′ m ′ /d ′ ij ≥ (1 − √ ε ) d ′ m ′ . So we can apply a Chernoff-type bound to see that there exists a constant c = c ( ε ) such that P ( | N G ′ ij ( v ) ∩ V ′ j | ≤ ( d ′ − √ ε ) m ′ ) ≤ e − cd ′ m ′ . Similarly, whenever X ⊆ V ′ i and Y ⊆ V ′ j are sets of size at least 2 εm ′ the expected numberof X - Y edges in G ′ ij is d G ′ ( X, Y ) d ′ | X || Y | /d ′ ij . Since ( V ′ i , V ′ j ) G ′ is 2 ε -regular this expectednumber lies between (1 − √ ε ) d ′ | X || Y | and (1 + √ ε ) d ′ | X || Y | . So again we can use a Chernoff-type bound to see that P ( | e G ′ ij ( X, Y ) − d ′ | X || Y || > √ ε | X || Y | ) ≤ e − cd ′ | X || Y | ≤ e − cd ′ ( εm ′ ) . Moreover, with probability at least 1 / (3 m ′ ) the graph G ′ ij has its expected density d ′ (seee.g. [4, p. 6]). Altogether this shows that with probability at least1 / (3 m ′ ) − m ′ e − cd ′ m ′ − m ′ e − cd ′ ( εm ′ ) > G ′ ij is ( √ ε, d ′ )-superregular and has density d ′ . Proceed similarly for every pairof clusters forming an edge of S . An analogous argument applied to a pair V i , V j of clusterswith V i V j ∈ E ( R ) \ E ( S ) shows that with non-zero probability the random subgraph G ′ ij is √ ε -regular and has density d ′ . Altogether this gives us the desired subgraph G ′ S of G ′ . (cid:3) Overview of the proof of Theorem 3.
Let G be our given oriented graph. Therough idea of the proof is to apply the Diregularity lemma and Lemma 8 to obtain a reducedoriented graph R and a pure oriented graph G ∗ . The following result of H¨aggkvist impliesthat R contains a 1-factor. Theorem 11 (H¨aggkvist [10]) . Let R be an oriented graph with δ ∗ ( R ) > (3 | R | − / .Then R is strongly connected and contains a -factor. So one can apply the Blow-up lemma (together with Proposition 10) to find a 1-factorin G ∗ − V ⊆ G − V . One now would like to glue the cycles of this 1-factor together andto incorporate the exceptional vertices to obtain a Hamilton cycle of G ∗ and thus of G .However, we were only able to find a method which incorporates a set of vertices whosesize is small compared to the cluster size m . This is not necessarily the case for V . So weproceed as follows. We first choose a random partition of the vertex set of G into two sets A and V ( G ) \ A having roughly equal size. We then apply the Diregularity lemma to G − A in order to obtain clusters V , . . . , V k and an exceptional set V . We let m denote the size of these clusters and set B := V ∪ . . . V k . By arguing as indicated above, we can find a Hamiltoncycle C B in G [ B ]. We then apply the Diregularity lemma to G − B , but with an ε which issmall compared to 1 /k , to obtain clusters V ′ , . . . , V ′ ℓ and an exceptional set V ′ . Since thechoice of our partition A, V ( G ) \ A will imply that δ ∗ ( G − B ) ≥ (3 / α/ | G − B | we can againargue as before to obtain a cycle C A which covers precisely the vertices in A ′ := V ′ ∪ · · · ∪ V ′ ℓ .Since we have chosen ε to be small compared to 1 /k , the set V ′ of exceptional vertices is nowsmall enough to be incorporated into our first cycle C B . (Actually, C B is only determinedat this point and not yet earlier on.) Moreover, by choosing C B and C A suitably we canensure that they can be joined together into the desired Hamilton cycle of G .4. Shifted Walks
In this section we will introduce the tools we need in order to glue certain cycles togetherand to incorporate the exceptional vertices. Let R ∗ be a digraph and let C be a collectionof disjoint cycles in R ∗ . We call a closed walk W in R ∗ balanced w.r.t. C if • for each cycle C ∈ C the walk W visits all the vertices on C an equal number oftimes, • W visits every vertex of R ∗ , • every vertex not in any cycle from C is visited exactly once.Let us now explain why balanced walks are helpful in order to incorporate the exceptionalvertices. Suppose that C is a 1-factor of the reduced oriented graph R and that R ∗ isobtained from R by adding all the exceptional vertices v ∈ V and adding an edge vV i (where V i is a cluster) whenever v sends edges to a significant proportion of the verticesin V i , say we add vV i whenever v sends at least cm edges to V i . (Recall that m denotes thesize of the clusters.) The edges in R ∗ of the form V i v are defined in a similar way. Let G c bethe oriented graph obtained from the pure oriented graph G ∗ by making all the non-emptybipartite subgraphs between the clusters complete (and orienting all the edges between theseclusters in the direction induced by R ) and adding the vertices in V as well as all the edgesof G between V and V ( G − V ). Suppose that W is a balanced closed walk in R ∗ whichvisits all the vertices lying on a cycle C ∈ C precisely m C ≤ m times. Furthermore, supposethat | V | ≤ cm/ V have distance at least 3 from each other on W .Then by ‘winding around’ each cycle C ∈ C precisely m − m C times (at the point when W first visits C ) we can obtain a Hamilton cycle in G c . Indeed, the two conditions on V ensurethat the neighbours of each v ∈ V on the Hamilton cycle can be chosen amongst the atleast cm neighbours of v in the neighbouring clusters of v on W in such a way that theyare distinct for different exceptional vertices. The idea then is to apply the Blow-up lemmato show that this Hamilton cycle corresponds to one in G . So our aim is to find such abalanced closed walk in R ∗ . However, as indicated in Section 3.2, the difficulties arisingwhen trying to ensure that the exceptional vertices lie on this walk will force us to applythe above argument to the subgraphs induced by a random partition of our given orientedgraph G .Let us now go back to the case when R ∗ is an arbitrary digraph and C is a collection ofdisjoint cycles in R ∗ . Given vertices a, b ∈ R ∗ , a shifted a - b walk is a walk of the form W = aa C b a C b . . . a t C t b t b where C , . . . , C t are (not necessarily distinct) cycles from C and a i is the successor of b i on C i for all i ≤ t . (We might have t = 0. So an edge ab is a shifted a - b walk.) Wecall C , . . . , C t the cycles which are traversed by W . So even if the cycles C , . . . , C t arenot distinct, we say that W traverses t cycles. Note that for every cycle C ∈ C the walk DIRAC TYPE RESULT ON HAMILTON CYCLES IN ORIENTED GRAPHS 11 W − { a, b } visits the vertices on C an equal number of times. Thus it will turn out that byjoining the cycles from C suitably via shifted walks and incorporating those vertices of R ∗ not covered by the cycles from C we can obtain a balanced closed walk on R ∗ .Our next lemma will be used to show that if R ∗ is oriented and δ ∗ ( R ∗ ) ≥ (3 / α ) | R ∗ | then any two vertices of R ∗ can be joined by a shifted walk traversing only a small numberof cycles from C (see Corollary 14). The lemma itself shows that the δ ∗ condition impliesexpansion, and this will give us the ‘expansion with respect to shifted neighbourhoods’we need for the existence of shifted walks. The proof of Lemma 12 is similar to that ofTheorem 11. Lemma 12.
Let R ∗ be an oriented graph on N vertices with δ ∗ ( R ∗ ) ≥ (3 / α ) N for some α > . If X ⊆ V ( R ∗ ) is nonempty and | X | ≤ (1 − α ) N then | N + ( X ) | ≥ | X | + αN/ . Proof.
For simplicity, we write δ := δ ( R ∗ ), δ + := δ + ( R ∗ ) and δ − := δ − ( R ∗ ). Suppose theassertion is false, i.e. there exists X ⊆ V ( R ∗ ) with | X | ≤ (1 − α ) N and(6) (cid:12)(cid:12) N + ( X ) (cid:12)(cid:12) < | X | + αN/ . We consider the following partition of V ( R ∗ ): A := X ∩ N + ( X ) , B := N + ( X ) \ X, C := V ( R ∗ ) \ ( X ∪ N + ( X )) , D := X \ N + ( X ) . (6) gives us(7) | D | + αN/ > | B | . Suppose A = ∅ . Then by an averaging argument there exists x ∈ A with | N + ( x ) ∩ A | < | A | /
2. Hence δ + ≤ | N + ( x ) | < | B | + | A | / . Combining this with (7) we get(8) | A | + | B | + | D | ≥ δ + − αN/ . If A = ∅ then N + ( X ) = B and so (7) implies | D | + αN/ ≥ | B | ≥ δ + . Thus (8) again holds.Similarly, if C = ∅ then considering the inneighbourhood of a suitable vertex x ∈ C gives(9) | B | + | C | + | D | ≥ δ − − αN/ . If C = ∅ then the fact that | X | ≤ (1 − α ) N and (6) together imply that D = ∅ . But then N − ( D ) ⊆ B and thus | B | ≥ δ − . Together with (7) this shows that (9) holds in this casetoo.If D = ∅ then trivially | A | + | B | + | C | = N ≥ δ . If not, then for any x ∈ D we have N ( x ) ∩ D = ∅ and hence(10) | A | + | B | + | C | ≥ | N ( x ) | ≥ δ. Combining (8), (9) and (10) gives3 | A | + 4 | B | + 3 | C | + 2 | D | ≥ δ − + 2 δ + + 2 δ − αN = 2 δ ∗ ( R ∗ ) − αN. Finally, substituting (7) gives3 N + αN/ ≥ δ ∗ ( R ∗ ) − αN ≥ N + αN, which is a contradiction. (cid:3) As indicated before, we will now use Lemma 12 to prove the existence of shifted walksin R ∗ traversing only a small number of cycles from a given 1-factor of R ∗ . For this (andlater on) the following fact will be useful. Fact 13.
Let G be an oriented graph with δ ∗ ( G ) ≥ (3 / α ) | G | for some constant α > .Then δ ( G ) > α | G | . Proof.
Suppose that δ − ( G ) ≤ α | G | . As G is oriented we have that δ + ( G ) < | G | / δ ∗ ( G ) < n/ α | G | , a contradiction. The proof for δ + ( G ) is similar. (cid:3) Corollary 14.
Let R ∗ be an oriented graph on N vertices with δ ∗ ( R ∗ ) ≥ (3 / α ) N forsome α > and let C be a -factor in R ∗ . Then for any distinct x, y ∈ V ( R ∗ ) there exists ashifted x - y walk traversing at most /α cycles from C . Proof.
Let X i be the set of vertices v for which there is a shifted x - v walk which traversesat most i cycles. So X = N + ( x ) = ∅ and X i +1 = N + ( X − i ) ∪ X i , where X − i is the set ofall predecessors of the vertices in X i on the cycles from C . Suppose that | X i | ≤ (1 − α ) N .Then Lemma 12 implies that | X i +1 | ≥ | N + ( X − i ) | ≥ | X − i | + αN/ | X i | + αN/ . So for i ∗ := ⌊ /α ⌋ −
1, we must have | X − i ∗ | = | X i ∗ | ≥ (1 − α ) N . But | N − ( y ) | ≥ δ − ( R ∗ ) > αN and so N − ( y ) ∩ X − i ∗ = ∅ . In other words, y ∈ N + ( X − i ∗ ) and so there is a shifted x - y walktraversing at most i ∗ + 1 cycles. (cid:3) Corollary 15.
Let R ∗ be an oriented graph with δ ∗ ( R ∗ ) ≥ (3 / α ) | R ∗ | for some < α ≤ / and let C be a -factor in R ∗ . Then R ∗ contains a closed walk which is balanced w.r.t. C and meets every vertex at most | R ∗ | /α times and traverses each edge lying on a cycle from C at least once. Proof.
Let C , . . . , C s be an arbitrary ordering of the cycles in C . For each cycle C i pick avertex c i ∈ C i . Denote by c + i the successor of c i on the cycle C i . Corollary 14 implies thatfor all i there exists a shifted c i - c + i +1 walk W i traversing at most 2 /α cycles from C , where c s +1 := c . Then the closed walk W ′ := c +1 C c W c +2 C c . . . W s − c + s C s c s W s c +1 is balanced w.r.t. C by the definition of shifted walks. Since each shifted walk W i traversesat most 2 /α cycles of C , the closed walk W meets each vertex at most ( | R ∗ | / /α ) + 1times. Let W denote the walk obtained from W ′ by ‘winding around’ each cycle C ∈ C once more. (That is, for each C ∈ C pick a vertex v on C and replace one of the occurencesof v on W ′ by vCv .) Then W is still balanced w.r.t. C , traverses each edge lying on a cyclefrom C at least once and visits each vertex of R ∗ at most ( | R ∗ | / /α ) + 2 ≤ | R ∗ | /α timesas required. (cid:3) Proof of Theorem 3
Partitioning G and applying the Diregularity lemma. Let G be an orientedgraph on n vertices with δ ∗ ( G ) ≥ (3 / α ) n for some constant α >
0. Clearly we mayassume that α ≪
1. Define positive constants ε, d and integers M ′ A , M ′ B such that1 /M ′ A ≪ /M ′ B ≪ ε ≪ d ≪ α ≪ . DIRAC TYPE RESULT ON HAMILTON CYCLES IN ORIENTED GRAPHS 13
Throughout this section, we will assume that n is sufficiently large compared to M ′ A for ourestimates to hold. Choose a subset A ⊆ V ( G ) with (1 / − ε ) n ≤ | A | ≤ (1 / ε ) n and suchthat every vertex x ∈ G satisfies d + ( x ) n − α ≤ | N + ( x ) ∩ A || A | ≤ d + ( x ) n + α N − ( x ) ∩ A satisfies a similar condition. (The existence of such a set A can be shown by considering a random partition of V ( G ).) Apply the Diregularity lemma(Lemma 7) with parameters ε , d + 8 ε and M ′ B to G − A to obtain a partition of the vertexset of G − A into k ≥ M ′ B clusters V , . . . , V k and an exceptional set V . Set B := V ∪ . . . ∪ V k and m B := | V | = · · · = | V k | . Let R B denote the reduced oriented graph obtained by anapplication of Lemma 8 and let G ∗ B be the pure oriented graph. Since δ + ( G − A ) / | G − A | ≥ δ + ( G ) /n − α/ A , Lemma 8 implies that(11) δ + ( R B ) ≥ ( δ + ( G ) /n − α/ | R B | . Similarly(12) δ − ( R B ) ≥ ( δ − ( G ) /n − α/ | R B | and δ ( R B ) ≥ ( δ ( G ) /n − α/ | R B | . Altogether this implies that(13) δ ∗ ( R B ) ≥ (3 / α/ | R B | . So Theorem 11 gives us a 1-factor C B of R B . We now apply Proposition 10 with C B playingthe role of S , ε playing the role of ε and d + 8 ε playing the role of d . This shows thatby adding at most 4 ε n further vertices to the exceptional set V we may assume that eachedge of R B corresponds to an ε -regular pair of density d (in the underlying graph of G ∗ B )and that each edge in the union S C ∈C B C ⊆ R B of all the cycles from C B corresponds toan ( ε, d )-superregular pair. (More formally, this means that we replace the clusters withthe subclusters given by Proposition 10 and replace G ∗ B with its oriented subgraph obtainedby deleting all edges not corresponding to edges of the graph G ′C B given by Proposition 10,i.e. the underlying graph of G ∗ B will now be G ′C B .) Note that the new exceptional set nowsatisfies | V | ≤ εn .Apply Corollary 15 with R ∗ := R B to find a closed walk W B in R B which is balancedw.r.t. C B , meets every cluster at most 2 | R B | /α times and traverses all the edges lying on acycle from C B at least once.Let G cB be the oriented graph obtained from G ∗ B by adding all the V i - V j edges for all thosepairs V i , V j of clusters with V i V j ∈ E ( R B ). Since 2 | R B | /α ≪ m B , we could make W B into aHamilton cycle of G cB by ‘winding around’ each cycle from C B a suitable number of times.We could then apply the Blow-up lemma to show that this Hamilton cycle corresponds toone in G ∗ B . However, as indicated in Section 3.2, we will argue slightly differently as it isnot clear how to incorporate all the exceptional vertices by the above approach.Set ε A := ε/ | R B | . Apply the Diregularity lemma with parameters ε A , d + 8 ε A and M ′ A to G [ A ∪ V ] to obtain a partition of the vertex set of G [ A ∪ V ] into ℓ ≥ M ′ A clusters V ′ , . . . , V ′ ℓ and an exceptional set V ′ . Let A ′ := V ′ ∪ · · · ∪ V ′ ℓ , let R A denote the reducedoriented graph obtained from Lemma 8 and let G ∗ A be the pure oriented graph. Similarlyas in (13), Lemma 8 implies that δ ∗ ( R A ) ≥ (3 / α/ | R A | and so, as before, we canapply Theorem 11 to find a 1-factor C A of R A . Then as before, Proposition 10 impliesthat by adding at most 4 ε A n further vertices to the exceptional set V ′ we may assume thateach edge of R A corresponds to an ε A -regular pair of density d and that each edge in the PSfrag replacements C C U − U U U +2 vW v Figure 2.
Incorporating the exceptional vertex v .union S C ∈C A C ⊆ R A of all the cycles from C A corresponds to an ( ε A , d )-superregular pair.So we now have that(14) | V ′ | ≤ ε A n = εn/ | R B | . Similarly as before, Corollary 15 gives us a closed walk W A in R A which is balanced w.r.t. C A ,meets every cluster at most 2 | R A | /α times and traverses all the edges lying on a cycle from C A at least once.5.2. Incorporating V ′ into the walk W B . Recall that the balanced closed walk W B in R B corresponds to a Hamilton cycle in G cB . Our next aim is to extend this walk to onewhich corresponds to a Hamilton cycle which also contains the vertices in V ′ . (The Blow-uplemma will imply that the latter Hamilton cycle corresponds to one in G [ B ∪ V ′ ].) We dothis by extending W B into a walk on a suitably defined digraph R ∗ B ⊇ R B with vertex set V ( R B ) ∪ V ′ in such a way that the new walk is balanced w.r.t. C B . R ∗ B is obtained fromthe union of R B and the set V ′ by adding an edge vV i between a vertex v ∈ V ′ and acluster V i ∈ V ( R B ) whenever (cid:12)(cid:12) N + G ( v ) ∩ V i (cid:12)(cid:12) > αm B /
10 and adding the edge V i v whenever (cid:12)(cid:12) N − G ( v ) ∩ V i (cid:12)(cid:12) > αm B /
10. Thus | N + G ( v ) ∩ B | ≤ | N + R ∗ B ( v ) | m B + | R B | αm B / . Hence | N + R ∗ B ( v ) | ≥ | N + G ( v ) ∩ B | /m B − α | R B | / ≥ | N + G ( v ) ∩ B || R B | / | B | − α | R B | / ≥ ( | N + G − A ( v ) | − | V | ) | R B | / | G − A | − α | R B | / ≥ ( δ + ( G ) /n − α/ | R B | ≥ α | R B | / . (15)(The penultimate inequality follows from the choice of A and the final one from Fact 13.)Similarly | N − R ∗ B ( v ) | ≥ α | R B | / . Given a vertex v ∈ V ′ pick U ∈ N + R ∗ B ( v ), U ∈ N − R ∗ B ( v ) \{ U } . Let C and C denote thecycles from C B containing U and U respectively. Let U − be the predecessor of U on C ,and U +2 be the successor of U on C . (15) implies that we can ensure U − = U +2 . (However,we may have C = C .) Corollary 14 gives us a shifted walk W v from U − to U +2 traversingat most 4 /α cycles of C B . To incorporate v into the walk W B , recall that W B traverses allthose edges of R B which lie on cycles from C B at least once. Replace one of the occurencesof U − U on W B with the walk W ′ v := U − W v U +2 C U vU C U , DIRAC TYPE RESULT ON HAMILTON CYCLES IN ORIENTED GRAPHS 15 i.e. the walk that goes from U − to U +2 along the shifted walk W v , it then winds oncearound C but stops in U , then it goes to v and further to U , and finally it winds around C .The walk obtained from W B by including v in this way is still balanced w.r.t. C B , i.e. eachvertex in R B is visited the same number of times as every other vertex lying on the samecycle from C B . We add the extra loop around C because when applying the Blow-up lemmawe will need the vertices in V ′ to be at a distance of at least 4 from each other. Using thisloop, this can be ensured as follows. After we have incorporated v into W B we ‘ban’ allthe 6 edges of (the new walk) W B whose endvertices both have distance at most 3 from v .The extra loop ensures that every edge in each cycle from C has at least one occurencein W B which is not banned. (Note that we do not have to add an extra loop around C since if C = C then all the banned edges of C lie on W ′ v but each edge of C also occurson the original walk W B .) Thus when incorporating the next exceptional vertex we canalways pick an occurence of an edge which is not banned to be replaced by a longer walk.(When incorporating v we picked U − U .) Repeating this argument, we can incorporate allthe exceptional vertices in V ′ into W B in such a way that all the vertices of V ′ have distanceat least 4 on the new walk W B .Recall that G cB denotes the oriented graph obtained from the pure oriented graph G ∗ B byadding all the V i - V j edges for all those pairs V i , V j of clusters with V i V j ∈ E ( R B ). Let G cB ∪ V ′ denote the graph obtained from G cB by adding all the V ′ - B edges of G as well as all the B - V ′ edges of G . Moreover, recall that the vertices in V ′ have distance at least 4 from eachother on W B and | V ′ | ≤ εn/ | R B | ≪ αm B /
20 by (14). As already observed at the beginningof Section 4, altogether this shows that by winding around each cycle from C B , one canobtain a Hamilton cycle C cB ∪ V ′ of G cB ∪ V ′ from the walk W B , provided that W B visits anycluster V i ∈ R B at most m B times. To see that the latter condition holds, recall that beforewe incorporated the exceptional vertices in V ′ into W B , each cluster was visited at most2 | R B | /α times. When incorporating an exceptional vertex we replaced an edge of W B by awalk whose interior visits every cluster at most 4 /α +2 ≤ /α times. Thus the final walk W B visits each cluster V i ∈ R B at most(16) 2 | R B | /α + 5 | V ′ | /α ( ) ≤ εn/ ( α | R B | ) ≤ √ εm B times. Hence we have the desired Hamilton cycle C cB ∪ V ′ of G cB ∪ V ′ . Note that (16) impliesthat we can choose C cB ∪ V ′ in such a way that for each cycle C ∈ C B there is subpath P C of C cB ∪ V ′ which winds around C at least(17) (1 − √ ε ) m B times in succession.5.3. Applying the Blow-up lemma to find a Hamilton cycle in G [ B ∪ V ′ ] . Ournext aim is to use the Blow-up lemma to show that C cB ∪ V ′ corresponds to a Hamiltoncycle in G [ B ∪ V ′ ]. Recall that k = | R B | and that for each exceptional vertex v ∈ V ′ theoutneighbour U of v on W B is distinct from its inneighbour U on W B . We will applythe Blow-up lemma with H being the underlying graph of C cB ∪ V ′ and G ∗ being the graphobtained from the underlying graph of G ∗ B by adding all the vertices v ∈ V ′ and joiningeach such v to all the vertices in N + G ( v ) ∩ U as well as to all the vertices in N − G ( v ) ∩ U .Recall that after applying the Diregularity lemma to obtain the clusters V , . . . , V k we usedProposition 10 to ensure that each edge of R B corresponds to an ε -regular pair of density d (in the underlying graph of G ∗ B and thus also in G ∗ ) and that each edge of the union S C ∈C B C ⊆ R B of all the cycles from C B corresponds to an ( ε, d )-superregular pair. V ′ will play the role of V in the Blow-up lemma and we take L , L , . . . , L k to be thepartition of H induced by V ′ , V , . . . , V k . φ : L → V ′ will be the obvious bijection (i.e. theidentity). To define the set I ⊆ V ( H ) of vertices of distance at least 4 from each otherwhich is used in the Blow-up lemma, let P ′ C be the subpath of H corresponding to P C (forall C ∈ C B ). For each i = 1 , . . . , k , let C i ∈ C B denote the cycle containing V i and let J i ⊆ L i consist of all those vertices in L i ∩ V ( P ′ C i ) which have distance at least 4 from theendvertices of P ′ C i . Thus in the graph H each vertex u ∈ J i has one of its neighbours in theset L − i corresponding to the predecessor of V i on C i and its other neighbour in the set L + i corresponding to the successor of V i on C i . Moreover, all the vertices in J i have distance atleast 4 from all the vertices in L and (17) implies that | J i | ≥ m B /
10. It is easy to see thatone can greedily choose a set I i ⊆ J i of size m B /
10 such that the vertices in S ki =1 I i havedistance at least 4 from each other. We take I := L ∪ S ki =1 I i .Let us now check conditions (C1)–(C9). (C1) holds with K := 1 since | L | = | V ′ | ≤ ε A n = εn/k ≤ d | H | . (C2) holds by definition of I . (C3) holds since H is a Hamilton cyclein G cB ∪ V ′ (c.f. the definition of the graph G cB ∪ V ′ ). This also implies that for every edge xy ∈ H with x ∈ L i , y ∈ L j ( i, j ≥
1) we must have that V i V j ∈ E ( R B ). Thus (C6) holds asevery edge of R B corresponds to an ε -regular pair of clusters having density d . (C4) holdswith K := 1 because | N H ( L ) ∩ L i | ≤ | L | = 2 (cid:12)(cid:12) V ′ (cid:12)(cid:12) ( ) ≤ εn/ | R B | ≤ εm B ≤ dm B . For (C5) we need to find a set D ⊆ I of buffer vertices. Pick any set D i ⊆ I i with | D i | = δ ′ m B and let D := S ki =1 D i . Since I i ⊆ J i we have that | N H ( D ) ∩ L j | = 2 δ ′ m B for all j = 1 , . . . , k .Hence || N H ( D ) ∩ L i | − | N H ( D ) ∩ L j || = 0for all 1 ≤ i < j ≤ k and so (C5) holds. (C7) holds with c := α/
10 by our choice U ∈ N + R ∗ B ( v )and U ∈ N − R ∗ B ( v ) of the neighbours of each vertex v ∈ V ′ in the walk W B (c.f. the definitionof the graph R ∗ B ).(C8) and (C9) are now the only conditions we need to check. Given a set E i ⊆ V i of sizeat most ε ′ m B , we wish to find F i ⊆ ( L i ∩ ( I \ D )) = I i \ D and a bijection φ i : E i → F i such that every v ∈ E i has a large number of neighbours in every cluster V j for which L j contains a neighbour of φ i ( v ). Pick any set F i ⊆ I i \ D of size | E i | . (This can be done since | D ∩ I i | = δ ′ m B and so | I i \ D | ≥ m B / − δ ′ m B ≫ ε ′ m B .) Let φ i : E i → F i be an arbitrarybijection. To see that (C8) holds with these choices, consider any vertex v ∈ E i ⊆ V i andlet j be such that L j contains a neighbour of φ i ( v ) in H . Since φ i ( v ) ∈ F i ⊆ I i ⊆ J i , thismeans that V j must be a neighbour of V i on the cycle C i ∈ C B containing V i . But thisimplies that | N G ∗ ( v ) ∩ V j | ≥ ( d − ε ) m B since each edge of the union S C ∈C B C ⊆ R B of allthe cycles from C B corresponds to an ( ε, d )-superregular pair in G ∗ .Finally, writing F := S ki =1 F i we have | N H ( F ) ∩ L i | ≤ ε ′ m B (since F j ⊆ J j for each j = 1 , . . . , k ) and so (C9) is satisfied with K := 2. Hence (C1)–(C9)hold and so we can apply the Blow-up lemma to obtain a Hamilton cycle in G ∗ such thatthe image of L i is V i for all i = 1 , . . . , k and the image of each x ∈ L is φ ( x ) ∈ V . (Recallthat G ∗ was obtained from the underlying graph of G ∗ B by adding all the vertices v ∈ V ′ and joining each such v to all the vertices in N + G ( v ) ∩ U as well as to all the vertices in N − G ( v ) ∩ U , where U and U are the neighbours of v on the walk W B .) Using the factthat H was obtained from the (directed) Hamilton cycle C cB ∪ V ′ and since U = U for each DIRAC TYPE RESULT ON HAMILTON CYCLES IN ORIENTED GRAPHS 17 v ∈ V ′ , it is easy to see that our Hamilton cycle in G ∗ corresponds to a (directed) Hamiltoncycle C B in G [ B ∪ V ′ ].5.4. Finding a Hamilton cycle in G . The last step of the proof is to find a Hamiltoncycle in G [ A ′ ] which can be connected with C B into a Hamilton cycle of G . Pick an arbitraryedge v v on C B and add an extra vertex v ∗ to G [ A ′ ] with outneighbourhood N + G ( v ) ∩ A ′ and inneighbourhood N − G ( v ) ∩ A ′ . A Hamilton cycle C A in the digraph thus obtainedfrom G [ A ′ ] can be extended to a Hamilton cycle of G by replacing v ∗ with v C B v . To findsuch a Hamilton cycle C A , we can argue as before. This time, there is only one exceptionalvertex, namely v ∗ , which we incorporate into the walk W A . Note that by our choice of A and B the analogue of (15) is satisfied and so this can be done as before. We then use theBlow-up lemma to obtain the desired Hamilton cycle C A corresponding to this walk.6. Proof of Theorem 4
The following observation guarantees that every oriented graph as in Theorem 4 has largeminimum semidegree.
Fact 16.
Suppose that < α < and that G is an oriented graph such that d + ( x ) + d − ( y ) ≥ (3 / α ) | G | whenever xy / ∈ E ( G ) . Then δ ( G ) ≥ | G | / α | G | / . Proof.
Suppose not. We may assume that δ + ( G ) ≤ δ − ( G ). Pick a vertex x with d + ( x ) = δ + ( G ). Let Y be the set of all those vertices y with xy / ∈ E ( G ). Thus | Y | ≥ | G | / − α | G | / d − ( y ) ≥ (3 / α ) | G | − d + ( x ) ≥ | G | / α | G | /
2. Hence e ( G ) ≥ | Y | (5 | G | / α | G | / > | G | /
64, a contradiction. (cid:3)
The proof of Theorem 4 is similar to that of Theorem 3. Fact 16 and Lemma 8 togetherimply that the reduced oriented graph R A (and similarly R B ) has minimum semidegreeat least | R | / G (i.e. it satisfies condition (d)of Lemma 8 with c = 3 / α ). Together with Lemma 17 below (which is an analogueof Lemma 12) this implies that R A (and R B as well) is an expander in the sense that | N + ( X ) | ≥ | X | + α | R A | / X ⊆ V ( R A ) with | X | ≤ (1 − α ) | R A | . In particular, R A (and similarly R B ) has a 1-factor: To see this, note that the above expansion propertytogether with Fact 16 imply that for any X ⊆ V ( R A ), we have | N + R A ( X ) | ≥ | X | . Togetherwith Hall’s theorem, this means that the following bipartite graph H has a perfect matching:the vertex classes W , W are 2 copies of V ( R A ) and we have an edge in H between w ∈ W and w ∈ W if there is an edge from w to w in R A . But clearly a perfect matching in H corresponds to a 1-factor in R A . Using these facts, one can now argue precisely as in theproof of Theorem 3. Lemma 17.
Suppose that < ε ≪ α ≪ . Let R ∗ be an oriented graph on N vertices andlet U be a set of at most εN ordered pairs of vertices of R ∗ . Suppose that d + ( x ) + d − ( y ) ≥ (3 / α ) N for all xy / ∈ E ( R ∗ ) ∪ U . Then any X ⊆ V ( R ∗ ) with αN ≤ | X | ≤ (1 − α ) N satisfies | N + ( X ) | ≥ | X | + αN/ . Proof.
The proof is similar to that of Lemma 12. Suppose that Lemma 17 does not holdand let X ⊆ V ( R ∗ ) with αN ≤ | X | ≤ (1 − α ) N be such that(18) (cid:12)(cid:12) N + ( X ) (cid:12)(cid:12) < | X | + αN/ . Call a vertex of R ∗ good if it lies in at most √ εN pairs from U . Thus all but at most 2 √ εN vertices of R ∗ are good. As in the proof of Lemma 12 we consider the following partition of V ( R ∗ ): A := X ∩ N + ( X ) , B := N + ( X ) \ X, C := V ( R ∗ ) \ ( X ∪ N + ( X )) , D := X \ N + ( X ) . (18) implies(19) | D | + αN/ > | B | . Suppose first that | D | > √ εN . It is easy to see that there are vertices x = y in D such that xy, yx / ∈ U . Since no edge of R ∗ lies within D we have xy, yx / ∈ E ( R ∗ ) and so d ( x ) + d ( y ) ≥ N/ αN . In particular, at least one of x, y has degree at least 3 N/ αN .But then(20) | A | + | B | + | C | ≥ N/ αN. If | D | ≤ √ εN then | A | + | B | + | C | ≥ N −| D | and so (20) still holds with room to spare. Notethat (19) and (20) together imply that 2 | A | +2 | C | ≥ N/ αN − | B | ≥ N/ −| B |−| D | ≥ N/
2. Thus at least one of
A, C must have size at least N/
8. In particular, this implies thatone of the following 3 cases holds.
Case 1. | A | , | C | > √ εN .Let A ′ be the set of all good vertices in A . By an averaging argument there exists x ∈ A ′ with | N + ( x ) ∩ A ′ | < | A ′ | /
2. Since N + ( A ) ⊆ A ∪ B this implies that | N + ( x ) | < | B | + | A \ A ′ | + | A ′ | /
2. Let C ′ ⊆ C be the set of all those vertices y ∈ C with xy / ∈ U . Thus | C \ C ′ | ≤ √ εN since x is good. By an averaging argument there exists y ∈ C ′ with | N − ( y ) ∩ C ′ | < | C ′ | /
2. But N − ( C ) ⊆ B ∪ C and so | N − ( y ) | < | B | + | C \ C ′ | + | C ′ | / d + ( x ) + d − ( y ) ≥ N/ αN since xy / ∈ E ( R ∗ ) ∪ U . Altogether this shows that | A ′ | / | C ′ | / | B | ≥ d + ( x ) + d − ( y ) − | A \ A ′ | − | C \ C ′ | ≥ N/ αN/ . Together with (20) this implies that 3 | A | + 6 | B | + 3 | C | ≥ N + 3 αN , which in turn togetherwith (19) yields 3 | A | + 3 | B | + 3 | C | + 3 | D | ≥ N + 3 αN/
2, a contradiction.
Case 2. | A | > √ εN and | C | ≤ √ εN .As in Case 1 we let A ′ be the set of all good vertices in A and pick x ∈ A ′ with | N + ( x ) | < | B | + | A \ A ′ | + | A ′ | /
2. Note that (19) implies that | D | > N − | X | − | C | − αN/ ≥ √ εN .Pick any y ∈ D such that xy / ∈ U . Then xy / ∈ E ( R ∗ ) since R ∗ contains no edges from A to D . Thus d + ( x ) + d − ( y ) ≥ N/ αN . Moreover, N − ( y ) ⊆ B ∪ C . Altogether this gives | A ′ | / | B | ≥ d + ( x ) + d − ( y ) − | A \ A ′ | − | C | ≥ N/ αN/ . As in Case 1 one can combine this with (20) and (19) to get a contradiction.
Case 3. | A | ≤ √ εN and | C | > √ εN .This time we let C ′ be the set of all good vertices in C and pick y ∈ C ′ with | N − ( y ) ∩ C ′ | < | C ′ | /
2. Hence | N − ( y ) | < | B | + | C \ C ′ | + | C ′ | /
2. Moreover, we must have | D | = | X | − | A | > √ εN . Pick any x ∈ D such that xy / ∈ U . Then xy / ∈ E ( R ∗ ) since R ∗ contains no edgesfrom D to C . Thus d + ( x ) + d − ( y ) ≥ N/ αN . Moreover, N + ( x ) ⊆ A ∪ B . Altogetherthis gives | C ′ | / | B | ≥ d + ( x ) + d − ( y ) − | A | − | C \ C ′ | ≥ N/ αN/ , which in turn yields a contradiction as before. (cid:3) Acknowledgement
We are grateful to Peter Keevash for pointing out an argument for the ‘shifted expansionproperty’ which is simpler than the one presented in an earlier version of this manuscript.
DIRAC TYPE RESULT ON HAMILTON CYCLES IN ORIENTED GRAPHS 19
References [1] N. Alon and A. Shapira, Testing subgraphs in directed graphs,
Journal of Computer and System Sci-ences (2004), 354–382.[2] N. Alon and J. Spencer, The Probabilistic Method (2nd edition), Wiley-Interscience 2000.[3] J. Bang-Jensen and G. Gutin,
Digraphs: Theory, Algorithms and Applications , Springer 2000.[4] B. Bollob´as,
Random Graphs (2nd edition), Cambridge University Press 2001.[5] B. Bollob´as and R. H¨aggkvist, Powers of Hamilton cycles in tournaments,
J. Combin. Theory B (1990), 309–318.[6] B. Csaba, On the Bollob´as–Eldridge conjecture for bipartite graphs, Combin. Probab. Comput. (2007),661–691.[7] B. Csaba, A. Shokoufandeh, and E. Szemer´edi, Proof of a conjecture of Bollob´as and Eldridge for graphsof maximum degree three, Combinatorica (2003), 35–72.[8] A. Frieze and M. Krivelevich, On packing Hamilton cycles in ε -regular graphs, J. Combin. Theory B (2005), 159–172.[9] A. Ghouila-Houri, Une condition suffisante d’existence d’un circuit hamiltonien, C.R. Acad. Sci. Paris (1960), 495–497.[10] R. H¨aggkvist, Hamilton cycles in oriented graphs, Combin. Probab. Comput. (1993), 25–32.[11] R. H¨aggkvist and A. Thomason, Oriented Hamilton cycles in oriented graphs, in Combinatorics, Geom-etry and Probability , Cambridge University Press 1997, 339–353.[12] P. Keevash, D. K¨uhn and D. Osthus, An exact minimum degree condition for Hamilton cycles in orientedgraphs, submitted.[13] L. Kelly, D. K¨uhn and D. Osthus, Cycles of given length in oriented graphs, submitted.[14] J. Koml´os, The Blow-up lemma,
Combin. Probab. Comput. (1999), 161–176.[15] J. Koml´os, G. N. S´ark¨ozy, and E. Szemer´edi, Blow-up lemma, Combinatorica (1997), 109–123.[16] J. Koml´os and M. Simonovits, Szemer´edi’s Regularity Lemma and its applications in graph theory, BolyaiSociety Mathematical Studies 2, Combinatorics, Paul Erd˝os is Eighty (Vol. 2) (D. Mikl´os, V. T. S´os andT. Sz˝onyi eds.), Budapest (1996), 295–352.[17] D. K¨uhn, D. Osthus and A. Treglown, Hamiltonian degree sequences in digraphs, submitted.[18] B. McKay, The asymptotic numbers of regular tournaments, Eulerian digraphs and Eulerian orientedgraphs,
Combinatorica (1990), 367–377.[19] O. Ore, Note on Hamilton circuits, American Math. Monthly (1960), 55.[20] C. Thomassen, Long cycles in digraphs with constraints on the degrees, in Surveys in Combinatorics (B. Bollob´as ed.),
London Math. Soc. Lecture Notes (1979), 211–228, Cambridge University Press.[21] C. Thomassen, Long cycles in digraphs. Proc. London Math. Soc. (1981), 231–251.[22] C. Thomassen, Edge-disjoint Hamiltonian paths and cycles in tournaments, Proc. London Math. Soc. (1982), 151–168.[23] D. Woodall, Sufficient conditions for cycles in digraphs, Proc. London Math Soc. (1972), 739–755.[24] A. Young, Extremal problems for dense graphs and digraphs, Master’s thesis, School of Mathematics,University of Birmingham 2005.Luke Kelly, Daniela K¨uhn & Deryk OsthusSchool of MathematicsUniversity of BirminghamEdgbastonBirminghamB15 2TTUK E-mail addresses: { kellyl,kuehn,osthus }}