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

In this paper, we establish an optimal $χ$-binding function for $(P_2\cup P_4,\text{ diamond})$-free graphs. We prove that for any graph $G$ in this class, $χ(G)\le 4$ when $ω(G)=2$, $χ(G)\le 6$ when $ω(G)=3$, and $χ(G)=ω(G)$ when $ω(G)\ge 4$, where $χ(G)$ and $ω(G)$ 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 $χ$-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.

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