Abstract
It is known that DP-coloring is a generalization of list coloring in simple graphs and many results in list coloring can be generalized in those of DP-coloring. In this work, we introduce a relaxed DP-\((k,d)^*\)-coloring which is a generalization of a \((k,d)^*\)-list coloring. We also show that every planar graph G without 4-cycles or 6-cycles is DP-\((3,1)^*\)-colorable. This generalizes the result of Lih et al. (Appl Math Lett 14(3):269–273, 2001) that such G is \((3,1)^*\)-choosable.
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References
Bernshteyn, A., Kostochka, A., Pron, S.: On DP-coloring of graphs and multigraphs. Sib. Math. J. 58(1), 28–36 (2017)
Chen, M., Raspaud, A.: On (3, 1)*-choosability of planar graphs without adjacent short cycles. Discrete Appl. Math. 162, 159–166 (2014)
Chen, M., Raspaud, A., Wang, W.: A (3, 1)*-choosable theorem on planar graphs. J. Comput. Optim. 32(3), 927–940 (2016)
Cowen, L., Cowen, R., Woodall, D.: Defective colorings of graphs in surfaces: partitions into subgraphs of bounded valency. J. Graph Theory 10(2), 187–195 (1986)
Dong, W., Xu, B.: A note on list improper coloring of plane graphs. Discrete Appl. Math. 157(2), 433–436 (2009)
Dvořák, Z., Postle, L.: Correspondence coloring and its application to list-coloring planar graphs without cycles of lengths 4 to 8. J. Combin. Theory Ser. B. 129, 38–54 (2018)
Eaton, E., Hull, T.: Defective list colorings of planar graphs. Bull. Inst. Combin. Appl. 25, 79–87 (1999)
Erdős P., Rubin A.L., Taylor H.: Choosability in graphs. In: Proceedings of West Coast Conference on Combinatorics, Graph Theory and Computing, Arcata, Sept. 5–7, Congr. Numer., vol. 26 (1979)
Kim, S.-J., Ozeki, K.: A sufficient condition for DP-\(4\)-colorability. Discrete Math. 341(7), 1983–1986 (2018)
Kim, S.-J., Yu, X.: Planar graphs without \(4\)-cycles adjacent to triangles are DP-\(4\)-colorable (2017). arXiv:1712.08999
Lih, K., Song, Z., Wang, W., Zhang, K.: A note on list improper coloring planar graphs. Appl. Math. Lett. 14(3), 269–273 (2001)
Škrekovski, R.: List improper colourings of planar graphs. Combin. Probab. Comput. 8(3), 293–299 (1999)
Thomassen, C.: Every planar graph is \(5\)-choosable. J. Combin. Theory Ser. B 62(1), 180–181 (1994)
Vizing, V.G.: Vertex colorings with given colors. Metody Diskret. Analiz. 29, 3–10 (1976)
Voigt, M.: List colourings of planar graphs. Discrete Math. 120(1–3), 215–219 (1993)
Wang, Y., Xu, L.: Improper choosability of planar graphs without 4-cycles. SIAM J. Discrete Math. 27(4), 2029–2037 (2013)
Acknowledgements
We would like to thank an anonymous referee for comments which are helpful for improvement of this paper. Also, the summarization part of referee’s report helps us to see another way to represent the concept of this work. Pongpat Sittitrai is supported by Development and Promotion of Science and Technology talents project (DPST). Kittikorn Nakprasit is supported by the Commission on Higher Education and the Thailand Research Fund under Grant RSA6180049.
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Sittitrai, P., Nakprasit, K. Sufficient Conditions on Planar Graphs to Have a Relaxed DP-3-Coloring. Graphs and Combinatorics 35, 837–845 (2019). https://doi.org/10.1007/s00373-019-02038-x
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DOI: https://doi.org/10.1007/s00373-019-02038-x