A note on a conjecture of Gy\'arf\'as

TL;DR

This paper proves that large enough triangle-free graphs with high chromatic number necessarily contain certain radius-three trees as induced subgraphs, confirming a specific case of Gyárfás's conjecture.

Contribution

It establishes that for a particular family of radius-three trees, any large enough triangle-free graph with high chromatic number contains an induced copy of these trees.

Findings

Large enough triangle-free graphs with high chromatic number contain specific radius-three trees.

Confirmed a case of Gyárfás's conjecture for a family of trees.

Provides conditions under which certain induced subgraphs must exist.

Abstract

This note proves that, given one member, $T$, of a particular family of radius-three trees, every radius-two, triangle-free graph, $G$, with large enough chromatic number contains an induced copy of $T$.