The asymptotic $\chi$-boundedness of hereditary families

TL;DR

This paper investigates the asymptotic chromatic bounds of hereditary graph families, showing that most graphs free of certain subgraphs have chromatic number equal to their clique number, with specific bounds for $C_6$-free graphs.

Contribution

It establishes that for various hereditary classes, almost all graphs are chromatically tight, and provides a new asymptotic bound for $C_6$-free graphs.

Findings

Almost all $T$-free graphs satisfy $ ext{chromatic number} = ext{clique number}$.

Almost all $C_k$-free graphs satisfy $ ext{chromatic number} = ext{clique number}$ for $k eq 6$.

The $C_6$-free graphs are asymptotically $ ext{chi}$-bounded with $f(w)=(1+o(1))w^2/ ext{log }w$.

Abstract

A family ${\cal F}$ of graphs is asymptotically $\chi$-bounded with bounding function $f$ if almost every graph $G$ in the family satisfies $\chi(G) \le f(\omega(G))$. A graph is $H$-free if it does not contain $H$ as an induced subgraph. We ask which hereditary families are asymptotically $\chi$-bounded, and discuss some related questions. We show that for every tree $T$, almost all $T$-free graphs $G$ satisfy $\chi(G)=\omega(G)$. We show that for every cycle $C_k$ except $C_6$, almost every $C_k$-free graph $G$ satisfies $\chi(G) = \omega(G)$. We show that the $C_6$-free graphs are asymptotically $\chi$-bounded with bounding function $f(w)=(1+o(1))\frac{w^2}{\log w}$.