Some results on chromatic number as a function of triangle count

TL;DR

This paper extends extremal bounds on the chromatic number to graphs with few triangles, providing tight bounds based on vertices, edges, and triangle counts, and discusses open problems related to fractional chromatic number.

Contribution

It introduces new bounds on the chromatic number as a function of vertices, edges, and triangle counts, interpolating between known bounds for all graphs and triangle-free graphs.

Findings

Bounds are tight for most cases.

Results interpolate between general and triangle-free graph bounds.

Open problem on fractional chromatic number may close remaining gaps.

Abstract

A variety of powerful extremal results have been shown for the chromatic number of triangle-free graphs. Three noteworthy bounds are in terms of the number of vertices, edges, and maximum degree given by Poljak \& Tuza (1994), and Johansson. There have been comparatively fewer works extending these types of bounds to graphs with a small number of triangles. One noteworthy exception is a result of Alon et. al (1999) bounding the chromatic number for graphs with low degree and few triangles per vertex; this bound is nearly the same as for triangle-free graphs. This type of parametrization is much less rigid, and has appeared in dozens of combinatorial constructions. In this paper, we show a similar type of result for $\chi(G)$ as a function of the number of vertices $n$, the number of edges $m$, as well as the triangle count (both local and global measures). Our results smoothly interpolate between the generic bounds true for all graphs and bounds for triangle-free graphs. Our results are tight for most of these cases; we show how an open problem regarding fractional chromatic number and degeneracy in triangle-free graphs can resolve the small remaining gap in our bounds.