Nearly optimal coloring of some C4-free graphs

TL;DR

This paper characterizes certain graph classes free of specific subgraphs and demonstrates they have polynomial-time algorithms for coloring with nearly optimal bounds, advancing understanding of graph coloring complexity.

Contribution

It identifies and analyzes $(C_3, C_4, F)$-free graphs without clique cutsets and universal cliques, establishing their $ ext{chi}$-polydet property with near-optimal bounds.

Findings

Characterization of $(C_3, C_4, F)$-free graphs without clique cutsets

Relation between $(C_4, F, H)$-free graphs and Petersen graph

Classes of $(C_4, F, H)$-free graphs are $ ext{chi}$-polydet with nearly optimal bounds

Abstract

A class ${\cal G}$ of graphs is $\chi$-{\em polydet} if ${\cal G}$ has a polynomial binding function $f$ and there is a polynomial time algorithm to determine an $f(\omega(G))$-coloring of $G\in {\cal G}$. Let $P_t$ and $C_t$ denote a path and a cycle on $t$ vertices, respectively. A {\em bull} consists of a triangle with two disjoint pendant edges, a {\em hammer} is obtained by identifying an end of $P_3$ with a vertex of a triangle, a {\em fork$^+$} is obtained from $K_{1, 3}$ by subdividing an edge twice. Let $H$ be a bull or a hammer, and $F$ be a $P_7$ or a fork$^+$. We determine all $(C_3, C_4, F)$-free graphs without clique cutsets and universal cliques, and present a close relation between $(C_4, F, H)$-free graphs and the Petersen graph. As a consequence, we show that the classes of $(C_4, F, H)$-free graphs are $\chi$-polydet with nearly optimal linear binding functions.