Excluding cycles with a fixed number of chords

TL;DR

This paper extends the characterization of graphs excluding cycles with a fixed number of chords, providing bounds on their chromatic number for cases with two and three chords.

Contribution

It generalizes previous results by establishing chromatic number bounds for graphs excluding cycles with exactly two or three chords.

Findings

Graphs with no cycle with exactly two chords have chromatic number at most 6.

Graphs with no cycle with exactly three chords have chromatic number at most max(96,w(G)+1).

Extends characterization of chord-excluding graphs to more complex cases.

Abstract

Trotignon and Vuskovic completely characterized graphs that do not contain cycles with exactly one chord. In particular, they show that such a graph G has chromatic number at most max(3,w(G)). We generalize this result to the class of graphs that do not contain cycles with exactly two chords and the class of graphs that do not contain cycles with exactly three chords. More precisely we prove that graphs with no cycle with exactly two chords have chromatic number at most 6. And a graph G with no cycle with exactly three chords have chromatic number at most max(96,w(G)+1).