A note on the Gy\'arf\'as-Sumner conjecture

TL;DR

This paper proves that for graphs avoiding large cliques and with high chromatic number, a subgraph similar to a given tree exists with a special path-induced property, advancing understanding of the Gyárfás-Sumner conjecture.

Contribution

It introduces the concept of a path-induced subgraph related to the Gyárfás-Sumner conjecture, providing a new partial result under the same hypotheses.

Findings

Existence of a path-induced subgraph isomorphic to the tree

Supports the Gyárfás-Sumner conjecture with a new structural property

Advances understanding of induced subgraph structures in high chromatic number graphs

Abstract

The Gy\'arf\'as-Sumner conjecture says that for every tree $T$ and every integer $t\ge 1$, if $G$ is a graph with no clique of size $t$ and with sufficiently large chromatic number, then $G$ contains an induced subgraph isomorphic to $T$. This remains open, but we prove that under the same hypotheses, $G$ contains a subgraph $H$ isomorphic to $T$ that is ``path-induced''; that is, for some distinguished vertex~$r$, every path of $H$ with one end $r$ is an induced path of $G$.