Suns in triangle-free graphs of large chromatic number

TL;DR

The paper proves that large chromatic number triangle-free graphs necessarily contain specific induced subgraphs called suns, extending previous open questions in graph theory.

Contribution

It establishes new bounds showing such graphs contain either large suns or nearly large suns as induced subgraphs.

Findings

Graphs with chromatic number at least 48 contain a t-sun for some t≥5 or a 4-sun minus one leaf.

For all ℓ≥5, sufficiently large chromatic number graphs contain a t-sun for some t≥ℓ or a 4-sun minus one leaf.

The results provide bounds linking chromatic number and specific induced subgraphs in triangle-free graphs.

Abstract

For an integer $t\geq 4$, a $t$-sun is a graph obtained from a $t$-vertex cycle $C$ by adding a degree-one neighbor for each vertex of $C$. Trotignon asked whether every triangle-free graph of sufficiently large chromatic number has an induced subgraph that is a $t$-sun for some $t\geq 4$. This remains open, but we show that every triangle-free graph of chromatic number at least $48$ has an induced subgraph that is either a $t$-sun for some $t\geq 5$, or a $4$-sun with a single degree-one vertex deleted. In fact, we prove that for all $\ell\geq 5$, there exists $c=c(\ell)\in \mathbb{N}$ such that every triangle-free graph of chromatic number at least $c$ has an induced subgraph that is either a $t$-sun for some $t\geq \ell$, or a $4$-sun with a single degree-one vertex deleted.