Suns in triangle-free graphs of large chromatic number

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.

Figures (1)

• Figure 1: Suns (top) versus what we exclude in \ref{['thm:maintrianglefreeshort']} (bottom).

Theorems & Definitions (25)

• Theorem 1.2
• Theorem 1.3
• Theorem 1.5: Hajebi bullfree
• Theorem 1.6
• Theorem 2.1
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Lemma 2.5