Critical $(P_5,W_4)$-Free Graphs

TL;DR

The authors study the class of $(P_5,W_4)$-free graphs with $k$-vertex-criticality, proving finiteness for every fixed $k$ and providing a complete characterization for $k=5$. They leverage the Strong Perfect Graph Theorem and a structural program around odd holes and antiholes, supported by two lemmas that analyze neighborhoods around a $C_5$ and around $igl( ext{overline}{C_7}igr)$. Their results yield a certifying polynomial-time algorithm for $k$-colorability in this graph family, where certificates are either a $k$-coloring or a $(k+1)$-vertex-critical induced subgraph; they also enumerate all 5-vertex-critical $(P_5,W_4)$-free graphs, finding 64 in one class and 21 in a tighter class, with the largest having 17 vertices. The work advances understanding of finiteness phenomena in vertex-critical graphs under $P_5$-free constraints and sets the stage for similar analyses on other five-vertex forbidden subgraphs.

Abstract

A graph $G$ is $k$-vertex-critical if $χ(G) = k$ but $χ(G-v)<k$ 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.

Figures (2)

• Figure 1: The $4$-wheel $W_4$.
• Figure 6: All 5-critical $(P_5,W_4)$-free graphs.

Theorems & Definitions (27)

• Theorem 1
• Theorem 2
• Lemma 1
• proof
• Lemma 2: CGHS21
• Lemma 3: XJGH23
• Theorem 3: BHS09MM12
• Theorem 4: The Strong Perfect Graph Theorem CRST06
• Lemma 4
• Lemma 5