Structure and colour in triangle-free graphs
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Motivated by a recent conjecture of the first author, we prove that every properly coloured triangle-free graph of chromatic number χ contains a rainbow independent set of size 1/2χ. This is sharp up to a factor 2. This result and its short proof have implications for the related notion of chromatic discrepancy. Drawing inspiration from both structural and extremal graph theory, we conjecture that every triangle-free graph of chromatic number χ contains an induced cycle of length Ω(χlogχ) as χ→∞. Even if one only demands an induced path of length Ω(χlogχ), the conclusion would be sharp up to a constant multiple. We prove it for regular girth 5 graphs and for girth 21 graphs. As a common strengthening of the induced paths form of this conjecture and of Johansson's theorem (1996), we posit the existence of some c >0 such that for every forest H on D vertices, every triangle-free and induced H-free graph has chromatic number at most c D/log D. We prove this assertion with `triangle-free' replaced by `regular girth 5'. In another direction, we provide an alternative, `bootstrapping' argument for a result of Kostochka, Sudakov and Verstraete (2017) that guarantees in any triangle-free graph of chromatic number χ a cycle of length Ω(χ^2logχ) as χ→∞.
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