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arXiv · 2511.13128

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

Hongyang Wang
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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.

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Hongyang Wang. 2025-11-17. Colouring ($P_2\cup P_4$, diamond)-free graphs with $\omega$ colours. https://arxiv.org/abs/2511.13128

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