Square-free graphs with no six-vertex induced path

TL;DR

This paper characterizes the structure of (P6,C4)-free graphs, providing tight bounds for their chromatic number and showing they have bounded clique-width, enabling polynomial-time algorithms for coloring and stability problems.

Contribution

It offers a structural characterization of (P6,C4)-free graphs and derives tight bounds for their chromatic number, also establishing bounded clique-width for certain subclasses.

Findings

Tight upper bounds for chromatic number in (P6,C4)-free graphs.

Structural decomposition into special classes or with clique cutsets.

Bounded clique-width for graphs with no clique cutset.

Abstract

We elucidate the structure of $(P_6,C_4)$-free graphs by showing that every such graph either has a clique cutset, or a universal vertex, or belongs to several special classes of graphs. Using this result, we show that for any $(P_6,C_4)$-free graph $G$, $\lceil\frac{5\omega(G)}{4}\rceil$ and $\lceil\frac{\Delta(G) + \omega(G) +1}{2}\rceil$ are tight upper bounds for the chromatic number of $G$. Moreover, our structural results imply that every ($P_6$,$C_4$)-free graph with no clique cutset has bounded clique-width, and thus the existence of a polynomial-time algorithm that computes the chromatic number (or stability number) of any $(P_6,C_4)$-free graph.