Optimal chromatic bound for ($P_2\cup P_4$, HVN)-free graphs

TL;DR

Problem: determine the χ-binding function for the class of $(P_2\cup P_4$, HVN)-free graphs. Approach: a structural decomposition using a maximal induced complete $\omega$-partite subgraph and a refined partition into perfect components, enabling a coloring strategy that yields a global bound. Main results: for $\omega(G)\ge4$, $\chi(G)\leq\lceil\frac{4}{3}\omega(G)\rceil$ with tightness shown by a $2\omega^2$-vertex construction; and exact bounds $\chi(G)\le4$ for $\omega(G)=2$ and $\chi(G)\le10$ for $\omega(G)=3$. Significance: extends and unifies several χ-bounding results for related forbidden-graph classes, providing an optimal, polynomial bound for this broad family.

Abstract

The HVN is a graph formed by removing two edges incident to the same vertex from the complete graph $K_5$. In this paper, we prove that every ($P_2\cup P_4$, HVN)-free graph $G$ satisfies $χ(G)\leq\lceil\frac{4}{3}ω(G)\rceil$ when $ω(G)\ge4$, where $χ(G)$ and $ω(G)$ denote the chromatic number and clique number of $G$, respectively. Furthermore, this bound is optimal for every $ω(G)\ge4$. Constructions demonstrating the optimality of the bound are provided. Our work unifies several previously known results on $χ$-binding functions for several graph classes.

Figures (1)

• Figure 1: The Mycielski--Grötzsch graph.

Theorems & Definitions (23)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• proof
• Lemma 3.1
• proof
• Lemma 3.2