Colouring ($P_2\cup P_4$, diamond)-free graphs with $\omega$ colours

TL;DR

This paper determines the exact chromatic bounds for $(P_2igcup P_4, ext{diamond})$-free graphs, extending previous results and providing optimal coloring functions based on clique number.

Contribution

It establishes the optimal $ ext{chi}$-binding function for $(P_2igcup P_4, ext{diamond})$-free graphs, refining and extending prior bounds.

Findings

$oxed{ ext{chi}(G) ext{ is at most }4 ext{ when } ext{omega}(G)=2}$

$oxed{ ext{chi}(G) ext{ is at most }6 ext{ when } ext{omega}(G)=3}$

$oxed{ ext{chi}(G)= ext{omega}(G) ext{ when } ext{omega}(G) ext{ is at least }4}$

Abstract

In this paper, we establish an optimal $\chi$-binding function for $(P_2\cup P_4,\text{ diamond})$-free graphs. We prove that for any graph $G$ in this class, $\chi(G)\le 4$ when $\omega(G)=2$, $\chi(G)\le 6$ when $\omega(G)=3$, and $\chi(G)=\omega(G)$ when $\omega(G)\ge 4$, where $\chi(G)$ and $\omega(G)$ denote the chromatic number and clique number of $G$, respectively. This result extends the known chromatic bounds for $(P_2\cup P_3,\text{ diamond})$-free graphs by showing that $(P_2\cup P_4,\text{ diamond})$-free graphs admit the same $\chi$-binding function. It also refines the chromatic bound obtained by Angeliya, Karthick and Huang [arXiv:2501.02543v3 [math.CO], 2025] for $(P_2\cup P_4,\text{ diamond})$-free graphs.