Induced subgraphs of graphs with large chromatic number. I. Odd holes

TL;DR

This paper proves Gyarfas's conjecture that graphs with no K_t subgraph and no odd holes are bounded in chromatic number, establishing a key link between forbidden subgraphs and graph coloring.

Contribution

We prove Gyarfas's conjecture, showing that graphs excluding both K_t and odd holes have bounded chromatic number, advancing understanding of graph coloring constraints.

Findings

Graphs with no K_t and no odd holes are n-colorable for some n.

The proof confirms the conjecture posed in 1985 by Gyarfas.

This result links forbidden subgraphs to chromatic bounds.

Abstract

An odd hole in a graph is an induced subgraph which is a cycle of odd length at least five. In 1985, A. Gyarfas made the conjecture that for all t there exists n such that every graph with no K_t subgraph and no odd hole is n-colourable. We prove this conjecture.