Linearly $\chi$-Bounding $(P_6,C_4)$-Free Graphs

TL;DR

This paper proves a linear bound on the chromatic number of $(P_6,C_4)$-free graphs, introduces a structure theorem for these graphs, and provides a polynomial-time approximation algorithm for coloring.

Contribution

It establishes a new linear $rac{3}{2}$-bound on chromatic number for $(P_6,C_4)$-free graphs and develops a polynomial-time coloring algorithm based on a novel structure theorem.

Findings

$oxed{ ext{Chromatic number} \\ ext{bounded by } rac{3}{2} imes ext{clique number}}$ for $(P_6,C_4)$-free graphs

A polynomial-time $3/2$-approximation algorithm for coloring these graphs

A structure theorem for $(P_6,C_4)$-free graphs without clique cutsets

Abstract

Given two graphs $H_1$ and $H_2$, a graph $G$ is $(H_1,H_2)$-free if it contains no subgraph isomorphic to $H_1$ or $H_2$. Let $P_t$ and $C_s$ be the path on $t$ vertices and the cycle on $s$ vertices, respectively. In this paper we show that for any $(P_6,C_4)$-free graph $G$ it holds that $\chi(G)\le \frac{3}{2}\omega(G)$, where $\chi(G)$ and $\omega(G)$ are the chromatic number and clique number of $G$, respectively. %Our bound is attained by $C_5$ and the Petersen graph. Our bound is attained by several graphs, for instance, the five-cycle, the Petersen graph, the Petersen graph with an additional universal vertex, and all $4$-critical $(P_6,C_4)$-free graphs other than $K_4$ (see \cite{HH17}). The new result unifies previously known results on the existence of linear $\chi$-binding functions for several graph classes. Our proof is based on a novel structure theorem on $(P_6,C_4)$-free graphs that do not contain clique cutsets. Using this structure theorem we also design a polynomial time $3/2$-approximation algorithm for coloring $(P_6,C_4)$-free graphs. Our algorithm computes a coloring with $\frac{3}{2}\omega(G)$ colors for any $(P_6,C_4)$-free graph $G$ in $O(n^2m)$ time.