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

TL;DR

This paper establishes an optimal upper bound on the chromatic number for a class of graphs defined by forbidden subgraphs, unifying and extending previous results in graph coloring theory.

Contribution

It proves a tight chromatic bound for ($P_2old P_4$, HVN)-free graphs, generalizing known results and providing constructions to demonstrate optimality.

Findings

The chromatic number is at most eil(4/3) times the clique number for the specified graph class.

The bound eil(4/3) imes ext{clique number} is proven to be optimal for all lique number  4.

The work unifies several existing results on inding functions for various graph classes.

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 $\chi(G)\leq\lceil\frac{4}{3}\omega(G)\rceil$ when $\omega(G)\ge4$, where $\chi(G)$ and $\omega(G)$ denote the chromatic number and clique number of $G$, respectively. Furthermore, this bound is optimal for every $\omega(G)\ge4$. Constructions demonstrating the optimality of the bound are provided. Our work unifies several previously known results on $\chi$-binding functions for several graph classes.