Complements of nearly perfect graphs

TL;DR

This paper investigates the chi-boundedness of graph classes and their complements, establishing bounds for certain classes and providing counterexamples for others, advancing understanding of graph complement properties.

Contribution

It proves that complements of certain chi-bounded classes are also chi-bounded, with explicit bounds, and constructs classes where this does not hold.

Findings

Complement of 3-chi-bounded class is chi-bounded by a linear function.

If a class is chi-bounded by a linear function beyond a threshold, its complement is also chi-bounded.

Counterexamples show some classes' complements are not chi-bounded.

Abstract

A class of graphs closed under taking induced subgraphs is $\chi$-bounded if there exists a function $f$ such that for all graphs $G$ in the class, $\chi(G) \leq f(\omega(G))$. We consider the following question initially studied in [A. Gy{\'a}rf{\'a}s, Problems from the world surrounding perfect graphs, {\em Zastowania Matematyki Applicationes Mathematicae}, 19:413--441, 1987]. For a $\chi$-bounded class $\cal C$, is the class $\bar{C}$ $\chi$-bounded (where $\bar{\cal C}$ is the class of graphs formed by the complements of graphs from $\cal C$)? We show that if $\cal C$ is $\chi$-bounded by the constant function $f(x)=3$, then $\bar{\cal C}$ is $\chi$-bounded by $g(x)=\lfloor\frac{8}{5}x\rfloor$ and this is best possible. We show that for every constant $c>0$, if $\cal C$ is $\chi$-bounded by a function $f$ such that $f(x)=x$ for $x \geq c$, then $\bar{\cal C}$ is $\chi$-bounded. For every $j$, we construct a class of graphs $\chi$-bounded by $f(x)=x+x/\log^j(x)$ whose complement is not $\chi$-bounded.