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

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.