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A Weak DP-Partitioning of Planar Graphs without 4-Cycles and 6-Cycles

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Abstract

In this paper, we introduce a weak DP-partitioning which combines the concepts of DP-coloring and vertex-partition. Let G be a planar graph without 4- and 6-cycles. We show that G is weakly DP-\(({\mathcal {F}}_2,{\mathcal {F}})\)-colorable. The result implies that G has a partition of the vertex set into two sets, where one set induces a forest, and the other induces a linear forest as shown by Huang et al. We also prove that G is weakly DP-\(({\mathcal {F}}_2,{\mathcal {F}}_0, {\mathcal {F}}_0)\)-colorable which improves the result by Fang and Wang that G is weakly DP-\(({\mathcal {F}}_{2},{\mathcal {F}}_{2}, {\mathcal {F}}_{0})\)-colorable.

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Acknowledgements

We would like to thank all referees for their instructive comments to improve this paper. This work has received scholarship under the Post-Doctoral Training Program from Khon Kaen University, Thailand (Grant No. PD2566-09). The third author was supported by National Research Council of Thailand (NRCT) [Grant number N41A640141].

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen, 40002, Thailand

    Pongpat Sittitrai, Keaitsuda Maneeruk Nakprasit & Kittikorn Nakprasit

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  1. Pongpat Sittitrai
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Sittitrai, P., Nakprasit, K.M. & Nakprasit, K. A Weak DP-Partitioning of Planar Graphs without 4-Cycles and 6-Cycles. Bull. Malays. Math. Sci. Soc. 46, 141 (2023). https://doi.org/10.1007/s40840-023-01528-9

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