Induced subgraphs of graphs with large chromatic number. II. Three steps towards Gyarfas' conjectures - Chudnovsky, M; Scott, A; Seymour, P (Elsevier, 2016)
Gyárfás conjectured in 1985 that for all k, ℓ, every graph with no clique of size more than k and no odd hole of length more than ℓ has chromatic number bounded by a function of k, ℓ. We prove three weaker statements:. •Every triangle-free graph with sufficiently large chromatic number has an odd hole of length different from five;•For all ℓ, every triangle-free graph with sufficiently large chromatic number contains either a 5-hole or an odd hole of length more than ℓ•For all k, ℓ, every graph with no clique of size more than k and sufficiently large chromatic number contains either a 5-hole or a hole of length more than ℓ.