On minimal triangle-free 6-chromatic graphs

TL;DR

This paper uses computational methods to determine bounds and classifications for minimal triangle-free graphs with specific chromatic numbers and girth, advancing understanding of graph coloring constraints.

Contribution

It provides the first bounds on the size of minimal triangle-free 6-chromatic graphs and classifies all triangle-free 5-chromatic graphs up to 24 vertices.

Findings

Smallest triangle-free 6-chromatic graphs have 32 to 40 vertices.

All triangle-free 5-chromatic graphs up to 24 vertices are classified.

Reed's conjecture holds for triangle-free graphs up to 24 vertices.

Abstract

A graph with chromatic number $k$ is called $k$-chromatic. Using computational methods, we show that the smallest triangle-free 6-chromatic graphs have at least 32 and at most 40 vertices. We also determine the complete set of all triangle-free 5-chromatic graphs up to 24 vertices. This implies that Reed's conjecture holds for triangle-free graphs up to at least this order. We also establish that the smallest regular triangle-free 5-chromatic graphs have 24 vertices. Finally, we show that the smallest 5-chromatic graphs of girth at least 5 have at least 29 vertices and that the smallest 4-chromatic graphs of girth at least 6 have at least 25 vertices.