Induced subgraphs of graphs with large chromatic number. XII. Distant stars

TL;DR

This paper advances the Gyarfas-Sumner conjecture by proving it for new families of trees with large chromatic number graphs, including double-ended brooms and two-legged caterpillars, expanding the classes of trees known to satisfy the conjecture.

Contribution

The paper proves the Gyarfas-Sumner conjecture for two new families of trees without the previous clustering restriction, including double-ended brooms and two-legged caterpillars.

Findings

Confirmed the conjecture for these new tree families.

Extended understanding of induced subgraphs in high chromatic number graphs.

Provided new techniques for analyzing trees with dispersed high-degree vertices.

Abstract

The Gyarfas-Sumner conjecture asserts that if H is a tree then every graph with bounded clique number and very large chromatic number contains H as an induced subgraph. This is still open, although it has been proved for a few simple families of trees, including trees of radius two, some special trees of radius three, and subdivided stars. These trees all have the property that their vertices of degree more than two are clustered quite closely together. In this paper, we prove the conjecture for two families of trees which do not have this restriction. As special cases, these families contain all double-ended brooms and two-legged caterpillars.