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Optimal chromatic bound for ($P_3\cup P_2$, house)-free graphs

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arxiv 2308.05442 v1 pith:JANHYKRF submitted 2023-08-10 math.CO

Rui Li , Di Wu , Jinfeng Li This is my paper · ORCID
classification math.CO
keywords housebounddenotefreegraphsomegaoptimalchromatic
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abstract

Let $G$ and $H$ be two vertex disjoint graphs. The {\em union} $G\cup H$ is the graph with $V(G\cup H)=V(G)\cup V(H)$ and $E(G\cup H)=E(G)\cup E(H)$. We use $P_k$ to denote a {\em path} on $k$ vertices, use {\em house} to denote the complement of $P_5$. In this paper, we show that $\chi(G)\le2\omega(G)$ if $G$ is ($P_3\cup P_2$, house)-free. Moreover, this bound is optimal when $\omega(G)\ge2$.

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  1. Coloring of some $(P_2\cup P_4)$-free graphs

    math.CO 2024-12 conditional novelty 6.0 of 10

    For (P2∪P4, gem)-free, (P2∪P4, butterfly)-free, and (P2∪P4, diamond)-free graphs, the paper establishes explicit χ-binding functions, and shows (P2∪P4, diamond, C5)-free graphs with clique number at least 5 are perfect.

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