Graphs with large minimum degree and no small odd cycles are $3$-colourable

TL;DR

This paper proves that large graphs with high minimum degree and no small odd cycles are 3-colorable, answering a question about graph coloring under specific structural constraints.

Contribution

It establishes a new threshold for minimum degree ensuring 3-colorability in graphs without small odd cycles, improving previous bounds.

Findings

Graphs with minimum degree at least .5n and no small odd cycles are 3-colorable.

The result applies to sufficiently large graphs, confirming conjectures about coloring in restricted graph classes.

A stronger version of the theorem holds with a slightly lower minimum degree threshold.

Abstract

Answering a question by Letzter and Snyder, we prove that for large enough $k$ any $n$-vertex graph $G$ with minimum degree at least $\frac{1}{2k-1}n$ and without odd cycles of length less than $2k+1$ is $3$-colourable. In fact, we prove a stronger result that works with a slightly smaller minimum degree.