Cycles in triangle-free graphs of large chromatic number
aa r X i v : . [ m a t h . C O ] A p r Cycles in triangle-free graphs of large chromatic number ∗ Alexandr Kostochka † Benny Sudakov ‡ Jacques Verstra¨ete § Abstract
More than twenty years ago Erd˝os conjectured [4] that a triangle-free graph G ofchromatic number k ≥ k ( ε ) contains cycles of at least k − ε different lengths as k → ∞ .In this paper, we prove the stronger fact that every triangle-free graph G of chromaticnumber k ≥ k ( ε ) contains cycles of ( − ε ) k log k consecutive lengths, and a cycle oflength at least ( − ε ) k log k . As there exist triangle-free graphs of chromatic number k with at most roughly 4 k log k vertices for large k , theses results are tight up to a constantfactor. We also give new lower bounds on the circumference and the number of differentcycle lengths for k -chromatic graphs in other monotone classes, in particular, for K r -freegraphs and graphs without odd cycles C s +1 . It is well-known that every k -chromatic graph has a cycle of length at least k for k ≥ k ≥ ⌊ ( k − ⌋ odd lengths. This is best possible in view of any graph whose blocks are complete graphs oforder k . Mihok and Schiermeyer [9] proved a similar result for even cycles: every graph G ofchromatic number k ≥ ⌊ k ⌋ − k ≥ ⌊ ( k − ⌋ consecutive lengths. Erd˝os [4] made the following conjecture: Conjecture 1.
For every ε > , there exists k ( ε ) such that for k ≥ k ( ε ) , every triangle-free k -chromatic graph contains more than k − ǫ odd cycles of different lengths. The second and third authors proved [12] that if G is a graph of average degree k and girth atleast five, then G contains cycles of Ω( k ) consecutive even lengths, and in [13] it was shownthat if an n -vertex graph of independence number at most nk is triangle-free, then it containscycles of Ω( k log k ) consecutive lengths. ∗ The authors thank Institute Mittag-Leffler (Djursholm, Sweden) for the hospitality and creative environ-ment. † Department of Mathematics, University of Illinois at Urbana–Champaign, IL, USA and Sobolev Instituteof Mathematics, Novosibirsk 630090, Russia, [email protected], Research supported in part by NSFgrant DMS-1266016 and by Grant NSh.1939.2014.1 of the President of Russia for Leading Scientific Schools. ‡ Department of Mathematics, ETH, 8092 Zurich. Email: [email protected]. Research sup-ported in part by SNSF grant 200021-149111 and by a USA-Israel BSF grant. § Department of Mathematics, University of California, San Diego, CA, USA, [email protected] supported in part by an Alfred P. Sloan Research Fellowship and NSF Grant DMS-0800704. .1 Main Result In this paper, we prove Conjecture 1 in the following stronger form:
Theorem 1.
For all ε > , there exists k ( ε ) such that for k ≥ k ( ε ) , every triangle-free k -chromatic graph G contains a cycle of length at least ( − ε ) k log k as well as cycles of atleast ( − ε ) k log k consecutive lengths. Kim [8] was the first to construct a triangle-free graph with chromatic number k and Θ( k log k )vertices. Bohman and Keevash [2] and Fiz Pontiveros, Griffiths and Morris [5] indepen-dently constructed a k -chromatic triangle-free graph with at most (4 + o (1)) k log k verticesas k → ∞ , refining the earlier construction of Kim [8]. These constructions show that thebound in Theorem 1 is tight up to a constant factor. Theorem 1 is a special case of a more general theorem on monotone properties. A graphproperty is called monotone if it holds for all subgraphs of a graph which has this property,i.e., is preserved under deletion of edges and vertices. Throughout this section, let n P ( k )denote the smallest possible order of a k -chromatic graph in a monotone property P . Definition 1.
Let α ≥ and let f : [3 , ∞ ) → R + . Then f is α -bounded if f is non-decreasingand whenever y ≥ x ≥ , y α f ( x ) ≥ x α f ( y ) . For instance, any polynomial with positive coefficients is α -bounded for some α ≥
1. We stressthat an α -bounded function is required to be a non-decreasing positive real-valued functionwith domain [3 , ∞ ). Theorem 2.
For all ε > and α, m ≥ , there exists k = k ( ε, α, m ) such that the followingholds. If P is a monotone property of graphs with n P ( k ) ≥ f ( k ) for k ≥ m and some α -bounded function f , then for k ≥ k , every k -chromatic graph G ∈ P contains ( i ) a cycle oflength at least (1 − ε ) f ( k ) and ( ii ) cycles of at least (1 − ε ) f ( k ) consecutive lengths. If n P ( k ) itself is α -bounded for some α , then we obtain from Theorem 2 a tight result thata k -chromatic graph in P contains a cycle of length asymptotic to n P ( k ) as k → ∞ . Butproving that n P ( k ) is α -bounded for some α is probably difficult for many properties, and inthe case P is the property of F -free graphs, perhaps is as difficult as obtaining asymptoticformulas for certain Ramsey numbers. Even in the case of the property of triangle-free graphs,we have seen n P ( k ) is known only up to a constant factor. We remark that in Theorems 1and 2, we have not attempted to optimize the quantities k ( ε ) and k ( ε, α, m ). K r -free graphs As an example of an application of Theorem 2, we consider the property P of K r -free graphs.A lower bound for the quantity n P ( k ) can be obtained by combining upper bounds for Ramsey2umbers together with a lemma on colorings obtained by removing maximum independentsets – see Section 5. In particular, we shall obtain the following from Theorem 2: Theorem 3. If G is a k -chromatic K r +1 -free graph, where r, k ≥ , then G contains a cycleof length Ω( k rr − ) , and cycles of Ω( k rr − ) consecutive lengths as k → ∞ . Theorem 3 is derived from upper bounds on the Ramsey numbers r ( K r , K t ) combined withTheorem 2. In general, if for a graph F one has r ( F, K t ) = O ( t a (log t ) − b ) for some a > b >
0, then any k -chromatic F -free graph has cycles ofΩ (cid:0) k aa − (log k ) ba − (cid:1) consecutive lengths. We omit the technical details, since the ideas of the proof are identicalto those used for Theorem 3. These technical details are presented in the proof of Theorem 1in Section 4, where F is a triangle (in which case a = 2 and b = 1), and the same ideas can beused to slightly improve Theorem 3 by logarithmic factors using better bounds on r ( K s , K t )from results of Ajtai, Koml´os and Szemer´edi [1]. Similarly, if C ℓ denotes the cycle of length ℓ , then it is known that r ( C s +1 , K t ) = O ( t /s (log t ) − /s ) – see [11]. This in turn providescycles of Ω( k s +1 log k ) consecutive lengths in any C s +1 -free k -chromatic graph, extendingTheorem 1. Notation and terminology.
For a graph G , let c ( G ) denote the length of a longest cyclein G and χ ( G ) the chromatic number of G . If F ⊂ G and S ⊂ V ( G ), let G [ F ] and G [ S ]respectively denote the subgraphs of G induced by V ( F ) and S . A chord of a cycle C in agraph is an edge of the graph joining two non-adjacent vertices on the cycle. All logarithmsin this paper are with the natural base. Organization.
In the next section, we present the lemmas which will be used to proveTheorem 2. Then in Section 4, we apply Theorem 2 to obtain the proof of Theorem 1.Theorem 3 is proved in Section 5. k -critical graphs When a small vertex cut is removed from a k -critical graph, all the resulting components stillhave relatively high chromatic number: Lemma 1.
Let G be a k -critical graph and let S be a vertex cut of G . Then for any component H of G − S , χ ( H ) ≥ k − | S | .Proof. If | S | + χ ( H ) ≤ k −
1, then a ( k − G − H (existing by the criticality of G ) can be extended to a ( k − G .3 .2 Nearly 3-connected subgraphs Our second lemma finds an almost 3-connected subgraph with high chromatic number in agraph with high chromatic number.
Lemma 2.
Let k ≥ . For every k -chromatic graph G , there is a graph G ∗ and an edge e ∗ ∈ E ( G ∗ ) such that (a) G ∗ − e ∗ ⊂ G and χ ( G ∗ − e ∗ ) ≥ k − . (b) G ∗ is -connected. (c) c ( G ∗ ) ≤ c ( G ) .Proof. Let G ′ be a k -critical subgraph of G . Then G ′ is 2-connected. If G ′ is 3-connected,then the lemma holds for G ∗ = G ′ with any e ∈ E ( G ′ ) as e ∗ . So suppose G ′ is not 3-connected.Among all separating sets S in G ′ of size 2 and components F of G ′ − S , choose a pair ( S, F )with the minimum | V ( F ) | . If S = { u, v } , then we let G ∗ be induced by V ( F ) ∪ S plus theedge e ∗ = uv . We claim χ ( G ∗ − e ∗ ) ≥ k −
1. Since G ′ is k -critical, there is a ( k − ϕ : V ( G ′ ) \ V ( F ) → { , , . . . , k − } of G ′ − V ( F ) and we may assume ϕ ( u ) = k − ϕ ( v ) ∈ { , k − } . Suppose for a contradiction that there is a coloring ϕ ∗ : V ( G ∗ − e ∗ ) → { , , . . . , k − } of G ∗ − e ∗ . If ϕ ( v ) = k −
1, then we let ϕ ′ ( x ) = ϕ ( x ) if x ∈ V ( G ′ ) − V ( F ) and ϕ ′ ( x ) = ϕ ∗ ( x ) if x ∈ V ( F ), and this ϕ ′ is a proper ( k − G ′ , a contradiction. Otherwise ϕ ( v ) = 1. Then we change the names of colors in ϕ ∗ so that ϕ ∗ ( v ) = 1 and again let ϕ ′ ( x ) = ϕ ( x ) if x ∈ V ( G ′ ) − V ( F ) and ϕ ′ ( x ) = ϕ ∗ ( x ) if x ∈ V ( F ).Again we have a proper ( k − G ′ . This contradiction proves (a).To prove (b), if G ∗ has a separating set S ′ with | S ′ | = 2, then, since uv ∈ E ( G ∗ ), it is alsoa separating set in G and at least one component of G ′ − S ′ is strictly contained in F . Thiscontradicts the choice of F and S .For (c), let C be a cycle in G ∗ with | C | = c ( G ∗ ). If e ∗ / ∈ E ( C ), then C is also a cycle in G , and thus c ( G ) ≥ | C | = c ( G ∗ ). If e ∗ ∈ E ( C ) and G ∗ = G ′ , then we obtain a longer cycle C in G ′ by replacing e ∗ with a uv -path in G ′ − V ( F ) – note such a path exists since G ′ is2-connected. This proves (c). In this subsection we show how to go from longest cycles in graphs to cycles of many consec-utive lengths. We will need the following result from [14], which is also implicit in the paperof Bondy and Simonovits [3]:
Lemma 3 (Lemma 2 in [14]) . Let H be a graph comprising a cycle with a chord. Let ( A, B )) be a nontrivial partition of V ( H ) . Then H contains A, B -paths of every positive length lessthan | H | , unless H is bipartite with bipartition ( A, B ) . emma 4. Let k ≥ and Q be a monotone class of graphs. Let h ( k, Q ) denote the smallestpossible length of a longest cycle in any k -chromatic graph in Q . Then every k -chromaticgraph in Q contains cycles of at least h ( k, Q ) consecutive lengths.Proof. Let F be a connected subgraph of G ∈ Q with chromatic number at least 4 k and let T be a breadth-first search tree in F . Let L i be the set of vertices at distance exactly i fromthe root of T in F . Then for some i , H = F [ L i ] has chromatic number at least 2 k . Let U be a breadth-first search tree in a component of H with chromatic number at least 2 k andlet M i be the set of vertices at distance exactly i from the root of U in H . Then for some i , J = H [ M i ] has chromatic number at least k . Let J ′ be a k -critical subgraph of J . Let P be a longest path in J ′ , so that | P | ≥ h ( k, Q ). Since J ′ has minimum degree at least k − ≥
3, each of the ends of P has at least two neighbors on P . In particular, there is apath P ′ ⊂ P of odd length with at least one chord, obtained by deleting at most one end of P , and | P ′ | ≥ h ( k, Q ) −
1. Then the ends of P ′ are joined by an even length path Q ⊂ U that is internally disjoint from P ′ , and C = Q ∪ P ′ is a cycle of odd length plus a chord, with | C | ≥ h ( k, Q ) + 1. Let ℓ := | C | and H ′ = G [ C ]. Now V ( H ′ ) ⊂ L i by construction. Let T ′ be a minimal subtree of T whose set of leaves is V ( H ′ ). Then T ′ branches at its root. Let A be the set of leaves in some branch of T ′ , and let B = V ( H ′ ) \ A . Then ( A, B ) is not abipartition of H ′ , since C has odd length, and therefore by Lemma 1 in [14], there exist paths P , P , . . . , P ℓ − ⊂ H ′ such that P i has length i and one end of P i is in A and one end of P i isin B , for i = 1 , , . . . , ℓ −
1. Now for each path P i , the ends of P i are joined by a path Q i oflength 2 r , where r is the height of T ′ and Q i and P i are internally disjoint. Therefore P i ∪ Q i is a cycle of length 2 r + i for i = 1 , , . . . , ℓ −
1, as required. α -bounded functions The following technical lemma is required for the proof of Theorem 2.
Lemma 5.
Let α, x ≥ , and let f be α -bounded. Then the function g ( x ) = xf ( x ) x + f ( x ) . is ( α + 1) -bounded, g ( x ) ≤ x for x ∈ [3 , x ] , and g ( x ) ≤ f ( x ) for all x ∈ [3 , ∞ ) .Proof. By definition, g ( x ) ≤ f ( x ) for x ∈ [3 , ∞ ) and g ( x ) ≤ x for x ∈ [3 , x ]. Also, since f is non-decreasing and positive on [3 , ∞ ), g is non-decreasing on [3 , ∞ ). It remains to checkthat g is ( α + 1)-bounded. For y ≥ x ≥
3, using that y α f ( x ) ≥ x α f ( y ), we find y α +1 g ( x ) = y α +1 xf ( x ) x + f ( x ) ≥ x α +1 yf ( y ) x + f ( x ) ≥ x α +1 g ( y ) . Therefore g is ( α + 1)-bounded. 5 Proof of Theorem 2
It is enough to prove Theorem 2 for all ε < /
2. Let β = α + 1, η = ε and x =max { m, ( βη ) β +1 } . Define k = k ( ε, α, m ) = ε f ( x ). Let g be a β -bounded functionin Lemma 5. We prove the following claim: Claim.
For k ≥ , every k -chromatic graph G ∈ P has a cycle of length at least (1 − η ) g ( k ) . Once this claim is proved, Theorem 2(i) follows since for k ≥ k ,(1 − η ) g ( k ) = (1 − ε ) kf ( k ) k + f ( x ) ≥ (1 − ε ) kf ( k ) k + εk/ ≥ (1 − ε ) f ( k ) , as required. Also, if k ≥ k , then by Lemma 4 every k -chromatic graph in P contains cyclesof at least (1 − η ) g ( k ) ≥ (1 − ε ) f ( k ) consecutive lengths, which gives Theorem 2(ii). Weprove the claim by induction on k ≥
3. For k ≤ x , g ( k ) ≤ k from Lemma 5, so in that case G contains a k -critical subgraph which has minimum degree at least k − k . This proves the claim for k ≤ x . Now suppose k > x . Let G ∗ be the graph obtained from G in Lemma 2. By Lemma 2(c), it is sufficient to show that G ∗ has a cycle of length at least (1 − η ) g ( k ). Let C be a longest cycle in G ∗ − e ∗ . By induction, | C | ≥ (1 − η ) g ( k − G = G ∗ [ C ] and χ = χ ( G ), and let G = G ∗ − G − e ∗ and χ := χ ( G ). Take C ′ to be a longest cycle in G . Let S be a minimum vertex set coveringall paths from C to C ′ . Either S separates C ′ − S from C − S or S = V ( C ′ ). Let | S | = ℓ .By Menger’s Theorem, G ∗ has ℓ vertex-disjoint paths P , P , . . . , P ℓ between C and C ′ – note ℓ ≥
3, as G ∗ is 3-connected. Let H = S ℓi =1 P i ∪ C ∪ C ′ . We find a cycle C ∗ ⊂ H with | C ∗ | ≥ ℓ − ℓ | C | + 12 | C ′ | . (1)To see this, first note that two of the paths, say P i and P j , contain ends at distance at most ℓ | C | on C , and now P i ∪ P j ∪ C ∪ C ′ contains a cycle C ∗ of length at least ℓ − ℓ | C | + 12 | C ′ | + | P i | + | P j | ≥ ℓ − ℓ | C | + 12 | C ′ | . At the same time, H contains a cycle C ∗∗ with | C ∗∗ | ≥
23 ( | C | + | C ′ | ) , (2)since there exist three cycles that together cover every edge of P ∪ P ∪ P ∪ C ∪ C ′ exactlytwice, and one of them has the required length. Now we complete the proof in three cases. Case 1. χ ≥ (1 − ηβ ) k . Then χ ≥ (1 − η ) k ≥ k ≥ m ≥
3, which implies n P ( χ ) ≥ f ( χ ).Since η ≤ < β , we have (1 − ηβ ) α ≥ − η . Since f is α -bounded, | C | ≥ n P ( χ ) ≥ f ( χ ) ≥ (1 − ηβ ) α f ( k ) ≥ (1 − η ) f ( k ) ≥ (1 − η ) g ( k ) . Case 2. χ < (1 − ηβ ) k and χ ≥ (1 − β ) k . Since χ ≥ g is β -bounded, g ( χ ) ≥ (1 − β ) β g ( k ) ≥ g ( k ) and g ( k − ≥ ( k − k ) β g ( k ) ≥ (1 − βk ) g ( k ) . k > x > β and | C ′ | ≥ (1 − η ) g ( χ ), we obtain from (2): | C ∗∗ | ≥ (1 − η ) g ( k −
1) + (1 − η ) g ( χ ) ≥ (1 − η ) g ( k ) · ( − βk ) > (1 − η ) g ( k ) . Case 3. χ < (1 − ηβ ) k and χ < (1 − β ) k . Then χ ≥ k − − χ > η β k ≥
3. Sinceevery χ -chromatic graph contains a cycle of length at least χ , we have that | C ′ | ≥ χ . If S = V ( C ′ ), then ℓ = | C ′ | ≥ χ > η β k . Otherwise, by Lemma 1, χ ≥ k − min { χ , ℓ } −
1, andso ℓ > k − χ − > η β k . By (1), and since g is β -bounded, | C ∗ | ≥ ℓ − ℓ (1 − η ) g ( k −
1) + g ( χ ) ≥ (1 − η ) g ( k ) · ( ℓ − ℓ ( k − k ) β + ( η β ) β ) ≥ (1 − η ) g ( k ) · (1 − βk − ℓ + ( η β ) β ) . Since k ≥ ( βη ) β +1 and ℓ > η β k , βk + ℓ ≤ βηk < ( η β ) β , and | C ∗ | > (1 − η ) g ( k ), as required. For the proof of Theorem 1, we use Theorem 2 with some specific choice of the function f ( k ).Let α ( G ) denote the independence number of a graph G . Shearer [10] showed the following: Lemma 6.
For every n -vertex triangle-free graph G with average degree d , α ( G ) > n log( d/e ) d . (3)This implies the following simple fact. In what follows, let ϕ ( x ) = ( x log x ) / . Lemma 7. If G is an n -vertex triangle-free graph and n ≥ e e , then α ( G ) ≥ ϕ ( n ) .Proof. Let d = ϕ ( n ). Then if n ≥ e e , d > e ( en ) / . Since G is triangle-free, the neighbor-hood of any vertex is an independent set, so we may assume G has maximum degree less than d . Then by (3) and since d > e ( en ) / , d > n log( d/e ) d > n log( en ) / ϕ ( n ) = ϕ ( n ) = d, a contradiction.To find a lower bound on the number of vertices of a triangle-free k -chromatic graph, werequire a lemma of Jensen and Toft [7] (they took s = 2, but their proof works for eachpositive integer s ): Lemma 8 ([7], Problem 7.3) . Let s ≥ and let ψ : [ s, ∞ ) → (0 , ∞ ) be a positive continuousnondecreasing function. Let P be a monotone class of graphs such that α ( G ) ≥ ψ ( | V ( G ) | ) forevery G ∈ P with | V ( G ) | ≥ s . Then for every such G with | V ( G ) | = n , χ ( G ) ≤ s + Z ns ψ ( x ) dx. emma 9. Let f ( x ) = cx log x where x ≥ and c > . Then f is 3-bounded.Proof. The function f is positive and non-decreasing, so one only has to check y f ( x ) ≥ x f ( y ) whenever y ≥ x ≥
3. This follows from y log x ≥ x log y for y ≥ x ≥
3, since thefunction y log y is increasing for y ≥ Lemma 10.
For every δ > , there exists k ( δ ) such that if k ≥ k ( δ ) , then every k -chromatictriangle-free graph has at least ( − δ ) k log k vertices.Proof. If δ ≥ the lemma is trivial, so suppose δ < . Let γ ( x ) = −
12 log ex . Let G be a k -chromatic triangle-free n -vertex graph. We apply the preceding lemma with ψ ( x ) = ϕ ( x )supplied by Lemma 7. For s ≥ e e , and using γ ( s ) ≤ γ ( x ) for x ≥ s : χ ( G ) ≤ s + √ Z ns ( x log ex ) − / dx = s + √ γ ( s ) Z ns ( x log ex ) − / γ ( s ) dx ≤ s + √ γ ( s ) Z ns ( x log x ) − / γ ( x ) dx. An antiderivative for the integrand is exactly x / (log ex ) − / , and therefore χ ( G ) ≤ s + √ γ ( s ) n / (log en ) − / . On the other hand, χ ( G ) ≥ k so if j = k − s , n ≥ γ ( s ) j log( γ ( s ) j ) . If s = ⌈ max { e e , e /δ }⌉ , then γ ( s ) ≥ (1 − δ ), so since δ < , n ≥ (1 − δ ) j log j ≥ (1 − δ ) j log j − k . If j ≥
3, then by Lemma 9, n ≥ (1 − δ ) ( jk ) k log k − k ≥ (1 − δ − sk ) k log k − k . Let k ( δ ) = s ≥ max { e e , e /δ } . Since k ≥ e /δ , k ≤ δk log k and 3 s ≤ δk . Therefore n ≥ (1 − δ − sk ) k log k − k ≥ (1 − δ ) k log k. This completes the proof.
Proof of Theorem 1.
The theorem is trivial if ε ≥ , so we assume ε < . We willderive Theorem 1 from Theorem 2. Let η = 2 ε, δ = ε . By Lemma 10, for k ≥ m := k ( δ ),every triangle-free k -chromatic graph G has at least ( − δ ) k log k vertices. Then f ( x ) =( − δ ) x log x is 3-bounded, by Lemma 9. By Theorem 2, with α = 3, G contains a cycleof length at least (1 − η ) f ( k ) as well as cycles of at least (1 − η ) f ( k ) consecutive lengthsin G , provided k ≥ k ( η, α, m ). Letting k ( ε ) = k ( η, α, m ) = k (2 ε, , k ( ε )), and noting(1 − η ) f ( k ) ≥ ( − ε ) k log k by the choice of η and δ , we have a cycle of length at least( − ε ) k log k in G whenever k ≥ k ( ε ). Similarly, if k is large enough relative to ε , then G contains cycles of at least (1 − η ) f ( k ) ≥ ( − ε ) k log k consecutive lengths. This completesthe proof. 8 Proof of Theorem 3
Lemma 11.
Let r ≥ and G be a K r -free n -vertex graph. Then α ( G ) ≥ n / ( r − − .Proof. If r ≥ n ≤ r − , then n / ( r − − ≤
1, so the claim holds. Let n > r − .If r = 3, either ∆( G ) ≥ n / or the graph is greedily ⌊ n / ⌋ + 1-colorable. Since vertexneighborhoods are independent sets in a triangle-free graph, either case gives an independentset of size at least n / −
1. For r >
3, either the graph has a vertex v of degree at least d ≥ n ( r − / ( r − , or the graph is n − / ( r − + 1-colorable. In the latter case, the largestcolor class is an independent set of size at least n / ( r − −
1. In the former case, since theneighborhood of v induces a K r − -free graph, by induction it contains an independent set ofsize at least d / ( r − − ≥ n / ( r − −
1, as required.
Lemma 12.
Let r ≥ and G be a K r -free n -vertex graph. Then χ ( G ) < n − / ( r − . Proof.
The function f ( x ) = max { , x / ( r − − } is positive continuous and nondecreasing.Since each nontrivial graph has an independent set of size 1, by Lemmas 8 and 11, χ ( G ) ≤ Z n f ( x ) d x ≤ Z n x / ( r − d x < n − / ( r − . Proof of Theorem 3.
We prove the first claim of the theorem for all k, r ≥
3, and then applyLemma 4 to prove the second claim. Let G be an n -vertex K r +1 -free graph with χ ( G ) = k ≥ k < n − r , so | V ( G ) | ≥ ( k ) rr − := f ( k ). Since f is rr − -bounded, the proof iscomplete by Theorem 2 with P the property of K r +1 -free graphs. • In this paper, we have shown that the length of a longest cycle and the length of a longestinterval of lengths of cycles in k -chromatic graphs G are large when G lacks certain subgraphs.In particular, when G has no triangles, this yields a proof of Conjecture 1 (in a stronger form).We believe that the following holds. Conjecture 2.
Let G be a k -chromatic triangle-free graph and let n k be the minimum numberof vertices in a k -chromatic triangle-free graph. Then G contains a cycle of length at least n k − o ( n k ) . • If Shearer’s bound [10] is tight, i.e., n k ∼ k log k , then the lower bound on the length ofthe longest cycle in any k -chromatic triangle-free graph in Theorem 1 would be tight.9 eferences [1] M. Ajtai, J. Koml´os and E. Szemer´edi, A note on Ramsey numbers, Journal of Combin.Theory Ser. A 29 (1980), 3, 354–360.[2] T. Bohman and P. Keevash, Dynamic concentration of the triangle-free process,http://arxiv.org/abs/1302.5963.[3] J. A. Bondy and M. Simonovits, Cycles of even length in graphs, J. Combin. Theory (B)16 (1974), 97–105.[4] P. Erd˝os, Some of my favourite problems in various branches of combinatorics, Matem-atiche (Catania) 47 (1992), 231–240.[5] G. Fiz Pontiveros, S. Griffiths and R. Morris, The triangle-free process and R (3 , k ),http://arxiv.org/abs/1302.6279.[6] A. Gy´arf´as, Graphs with k odd cycle lengths. Discrete Mathematics 103(1) (1992), 41–48.[7] T. R. Jensen and B. Toft, Graph Coloring Problems, Wiley-Interscience Series in DiscreteMathematics and Optimization, John Wiley & Sons, New York, 1995.[8] J. Kim, The Ramsey number R (3; t ) has order of magnitude t / log tt