Sufficient Conditions on Planar Graphs to Have a Relaxed DP-3-Coloring

  • Pongpat Sittitrai
  • Kittikorn NakprasitEmail author
Original Paper


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.


DP-coloring Improper colorings Planar graphs 

Mathematics Subject Classification




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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Copyright information

© Springer Japan KK, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceKhon Kaen UniversityKhon KaenThailand

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