Induced Cycles and Chromatic Number

We prove that, for any pair of integers k, l ≥ 1, there exists an integerN(k, l) such that every graph with chromatic number at least N(k, l) contains either Kk or an induced odd cycle of length at least 5 or an induced cycle of length at least l. Given a graph with large chromatic number, it is natural to ask whether it must contain induced subgraphs with particular properties. One possibility is that the graph contains a large clique. If this is not the case, however, are there other graphs that G must then contain as induced subgraphs? For instance, given H and k, does every graph of sufficiently large chromatic number contain either Kk or an induced copy of H? Of course, there are graphs with arbitrarily large chromatic number and girth, so H must be acyclic. Gyárfás [1] and Sumner [8] independently made the beautiful (and difficult) conjecture that for every tree T and integer k there is an integer f(k, T ) such that every graph G with χ(G) ≥ f(k, T ) contains either Kk or an induced copy of T . Kierstead and Penrice [4], extending a result of Gyárfás, Szemerédi and Tuza [3], have proved the Gyárfás-Sumner conjecture for trees of radius at most two, while Kierstead, Penrice and Trotter [5] have resolved the on-line version of the conjecture. Scott [7] proved a ‘topological’ version of the conjecture: for every tree T and integer k there is an integer f(k, T ) such that every graph G with χ(G) ≥ f(k, T ) contains either Kk or an induced copy of a subdivision of T . It is also known that for any tree T and integer k, every graph of sufficiently large chromatic number contains either an induced copy of T , an induced Kk,k or Kk, see Kierstead and Rödl [6]. Another conjecture relating induced subgraphs to chromatic number is the well-known Strong Perfect Graph Conjecture, which asserts that every graph G such that neither G nor G contains an induced odd cycle of length at least 5 must satisfy χ(G) = cl(G). In other words, if χ(G) ≥ cl(G) then G must contain either Kk, or an induced odd cycle of length at least 5, or the complement of such a cycle.