$\chi$-bounds, operations and chords

TL;DR

This paper characterizes long-unichord-free graphs, provides an efficient recognition algorithm, and demonstrates they can be colored with a polynomial bound related to their clique number.

Contribution

It introduces a structure theorem for long-unichord-free graphs, offers a recognition algorithm, and establishes a polynomial coloring bound.

Findings

Structure theorem for long-unichord-free graphs

Recognition algorithm with O(n^4m) complexity

Coloring bound of O(ω^3) for such graphs

Abstract

A \emph{long unichord} in a graph is an edge that is the unique chord of some cycle of length at least 5. A graph is \emph{long-unichord-free} if it does not contain any long-unichord. We prove a structure theorem for long-unichord-free graph. We give an $O(n^4m)$-time algorithm to recognize them. We show that any long-unichord-free graph $G$ can be colored with at most $O(\omega^3)$ colors, where $\omega$ is the maximum number of pairwise adjacent vertices in $G$.