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A Tight Linear Bound to the Chromatic Number of \((P_5, K_1+(K_1\cup K_3))\)-Free Graphs

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Abstract

Let \(F_1\) and \(F_2\) be two disjoint graphs. The union \(F_1\cup F_2\) is a graph with vertex set \(V(F_1)\cup V(F_2)\) and edge set \(E(F_1)\cup E(F_2)\), and the join \(F_1+F_2\) is a graph with vertex set \(V(F_1)\cup V(F_2)\) and edge set \(E(F_1)\cup E(F_2)\cup \{xy\;|\; x\in V(F_1)\hbox { and } y\in V(F_2)\}\). In this paper, we present a characterization to \((P_5, K_1\cup K_3)\)-free graphs, prove that \(\chi (G)\le 2\omega (G)-1\) if G is \((P_5, K_1\cup K_3)\)-free. Based on this result, we further prove that \(\chi (G)\le \)max\(\{2\omega (G),15\}\) if G is a \((P_5,K_1+(K_1\cup K_3))\)-free graph. We also construct a \((P_5, K_1+( K_1\cup K_3))\)-free graph G with \(\chi (G)=2\omega (G)\).

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Acknowledgements

We thank the anonymous reviewers for valuable comments, and thank Dr. Karthick for pointing out an error in our earlier version on the construction of some extremal graphs.

Funding

This work was supported by National Natural Science Foundation of China (No. 11931106 and 12101117) and by Natural Science Foundation of Jiangsu Province (No. BK20200344). Author Baogang Xu has received research support from National Natural Science Foundation of China. Author Yian Xu has received research support from National Natural Science Foundation of China and Natural Science Foundation of Jiangsu Province.

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Authors and Affiliations

  1. School of Information and Engineering, Nanjing Xiaozhuang University, Nanjing, 211171, China

    Wei Dong

  2. Institute of Mathematics, School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210023, China

    Baogang Xu

  3. School of Mathematics, Southeast University, 2 SEU Road, Nanjing, 211189, China

    Yian Xu

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  1. Wei Dong
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  2. Baogang Xu
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  3. Yian Xu
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Contributions

All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by WD, BX and YX. The first draft of the manuscript was written by WD, BX and YX, and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

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Correspondence to Yian Xu.

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Supported by NSFC No. 11931106 and 12101117, and by NSFJS No. BK20200344.

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Dong, W., Xu, B. & Xu, Y. A Tight Linear Bound to the Chromatic Number of \((P_5, K_1+(K_1\cup K_3))\)-Free Graphs. Graphs and Combinatorics 39, 43 (2023). https://doi.org/10.1007/s00373-023-02642-y

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  • DOI: https://doi.org/10.1007/s00373-023-02642-y

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