Structure and colour in triangle-free graphs

TL;DR

This paper proves new bounds on the chromatic number of triangle-free graphs with proper colorings, showing the existence of large rainbow independent sets and conjecturing the presence of long induced cycles, with partial results supporting these ideas.

Contribution

It establishes a sharp bound on rainbow independent sets in properly colored triangle-free graphs and proposes a conjecture linking chromatic number and induced cycles, supported by partial proofs.

Findings

Every properly colored triangle-free graph of chromatic number χ contains a rainbow independent set of size ⌈χ/2⌉.

Conjecture: triangle-free graphs of chromatic number χ contain induced cycles of length Ω(χ log χ).

Proved the conjecture for regular girth 5 and girth 21 graphs.

Abstract

Motivated by a recent conjecture of the first author, we prove that every properly coloured triangle-free graph of chromatic number $\chi$ contains a rainbow independent set of size $\lceil\frac12\chi\rceil$. 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 $\chi$ contains an induced cycle of length $\Omega(\chi\log\chi)$ as $\chi\to\infty$. Even if one only demands an induced path of length $\Omega(\chi\log\chi)$, 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 $5$'.