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

TL;DR

This paper determines the exact χ-binding function for the class of $P_2\cup P_4$-free, diamond-free graphs. A structural decomposition around a maximum clique yields partitions and inter-block properties that separate perfect components from challenging regions and enable efficient colourings. The main results give tight bounds: $χ(G)≤4$ for $ω(G)=2$, $χ(G)≤6$ for $ω(G)=3$, and $χ(G)=ω(G)$ for $ω(G)≥4$, with the caveat that some graphs with $ω(G)≥4$ are not perfect; notably these bounds align with those for $P_2\cup P_3$-free, diamond-free graphs. The work thus completes the χ-binding function for this graph class and provides a decomposition-based framework applicable to related forbidden-subgraph families.

Abstract

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.

Figures (6)

• Figure 1: An imperfect ($P_2 \cup P_4$, diamond)-free graph $G$ with $\chi(G)=\omega(G)=5$.
• Figure 2: Two cases in the proof of Lemma \ref{['L31']}.
• Figure 3: The case $|X\cap C_0|=2$ in the proof of Lemma \ref{['L32']}. Here a dashed line indicates that the two vertices may or may not be adjacent.
• Figure 4: The case $|X\cap C_0|=1$ in the proof of Lemma \ref{['L32']}.
• Figure 5: Two cases in the proof of Lemma \ref{['L33']}.

Theorems & Definitions (33)

• Conjecture 1.1
• Lemma 1.2
• Theorem 1.3
• Lemma 1.4
• proof
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3