Critical $(P_5,W_4)$-Free Graphs

TL;DR

This paper proves the finiteness and characterizes all 5-vertex-critical graphs that are free of certain subgraphs, leading to efficient algorithms for k-colorability in these graph classes.

Contribution

It establishes the finiteness of k-vertex-critical (P5,W4)-free graphs for all k and characterizes the 5-vertex-critical cases, enabling polynomial-time colorability algorithms.

Findings

Finiteness of k-vertex-critical (P5,W4)-free graphs for all k

Complete characterization of 5-vertex-critical (P5,W4)-free graphs

Existence of polynomial-time certifying algorithms for k-colorability

Abstract

A graph $G$ is $k$-vertex-critical if $\chi(G) = k$ but $\chi(G-v)<k$ for all $v \in V(G)$. A graph is $(H_1,H_2)$-free if it contains no induced subgraph isomorphic to $H_1$ nor $H_2$. A $W_4$ is the graph consisting of a $C_4$ plus an additional vertex adjacent to all the vertices of the $C_4$. We show that there are finitely many $k$-vertex-critical $(P_5,W_4)$-free graphs for all $k \ge 1$ and we characterize all $5$-vertex-critical $(P_5,W_4)$-free graphs. Our results imply the existence of a polynomial-time certifying algorithm to decide the $k$-colorability of $(P_5,W_4)$-free graphs for each $k \ge 1$ where the certificate is either a $k$-coloring or a $(k+1)$-vertex-critical induced subgraph.