Chi-boundedness of graphs containing no cycles with $k$ chords

TL;DR

This paper proves that graphs without cycles having exactly k chords are chi-bounded for large k or specific forms, confirming a conjecture for most values of k.

Contribution

It establishes chi-boundedness for graphs excluding cycles with exactly k chords, verifying a conjecture for all but finitely many k.

Findings

Graphs with no cycle with exactly k chords are chi-bounded for large or specific k.

The result confirms a conjecture of Aboulker and Bousquet for almost all k.

Provides new insights into the structure of chordless cycles in graphs.

Abstract

We prove that the family of graphs containing no cycle with exactly $k$-chords is $\chi$-bounded, for $k$ large enough or of form $\ell(\ell-2)$ with $\ell \ge 3$ an integer. This verifies (up to a finite number of values $k$) a conjecture of Aboulker and Bousquet (2015).