Classes of graphs with no long cycle as a vertex-minor are polynomially $\chi$-bounded

TL;DR

This paper proves that graphs excluding long cycles as vertex-minors are polynomially chi-bounded, establishing a polynomial bound on their chromatic number based on clique number, and analyzing closure properties under certain graph operations.

Contribution

It demonstrates that classes of graphs with no long cycle vertex-minor are polynomially chi-bounded and explores how this property is preserved under the 1-join operation.

Findings

Existence of a polynomial chi-bounding function for graphs excluding long cycle vertex-minors.

Closure of polynomial chi-boundedness under the 1-join operation.

Polynomial bounds depend only on the cycle length $n$.

Abstract

A class $\mathcal G$ of graphs is $\chi$-bounded if there is a function $f$ such that for every graph $G\in \mathcal G$ and every induced subgraph $H$ of $G$, $\chi(H)\le f(\omega(H))$. In addition, we say that $\mathcal G$ is polynomially $\chi$-bounded if $f$ can be taken as a polynomial function. We prove that for every integer $n\ge3$, there exists a polynomial $f$ such that $\chi(G)\le f(\omega(G))$ for all graphs with no vertex-minor isomorphic to the cycle graph $C_n$. To prove this, we show that if $\mathcal G$ is polynomially $\chi$-bounded, then so is the closure of $\mathcal G$ under taking the $1$-join operation.