Every Planar Graph Without Pairwise Adjacent 3-, 4-, and 5-Cycle is DP-4-Colorable

  • Pongpat Sittitrai
  • Kittikorn NakprasitEmail author


DP-coloring is a generalization of list coloring in simple graphs. Many results in list coloring can be generalized in those of DP-coloring. Kim and Ozeki showed that every planar graph without k-cycles where \(k=3,4,5,\) or 6 is DP-4-colorable. Recently, Kim and Yu extended the result on 3- and 4-cycles by showing that every planar graph without triangles adjacent to 4-cycles are DP-4-colorable. Xu and Wu showed that every planar graph without 5-cycles adjacent simultaneously to 3-cycles and 4-cycles is 4-choosable. In this paper, we extend the results on 3-, 4-, and 5-cycles as follows. Let G be a planar graph without pairwise adjacent 3-, 4-, and 5-cycle. We prove that each precoloring of a 3-cycle of G can be extended to a DP-4-coloring of G. As a consequence, each planar graph without pairwise adjacent 3-, 4-, and 5-cycle is DP-4-colorable.


DP-coloring List coloring Planar graph Cycle 

MSC code




We would like to thank anonymous referees for comments which are helpful for improvement in this paper. The first author is supported by Development and Promotion of Science and Technology talents project (DPST). The second author is supported by the Commission on Higher Education and the Thailand Research Fund under Grant RSA6180049.


  1. 1.
    Alon, N.: Degrees and choice numbers. Random Struct. Algorithms 16, 364–368 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bernshteyn, A.: The asymptotic behavior of the correspondence chromatic number. Discrete Math. 339, 2680–2692 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bernshteyn, A., Kostochka, A., Pron, S.: On DP-coloring of graphs and multigraphs. Sib. Math. J 58, 28–36 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Dvořák, Z., Postle, L.: Correspondence coloring and its application to list-coloring planar graphs without cycles of lengths \(4\) to \(8\). J. Comb. Theory Ser. B 129, 38–54 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Erdős, P., Rubin, A.L., Taylor, H.: Choosability in graphs. In: Proceedings, West Coast Conference on Combinatorics, Graph Theory and Computing, Arcata, CA., Sept. 5–7, Congressus Numerantium, vol. 26 (1979)Google Scholar
  6. 6.
    Kim, S.-J., Ozeki, K.: A sufficient condition for DP-\(4\)-colorability. Discrete Math. 341, 1983–1986 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Kim, S.-J., Yu, X.: Planar graphs without \(4\)-cycles adjacent to triangles are DP-\(4\)-colorable. Graphs Comb. 35(3), 707–718 (2019)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Thomassen, C.: Every planar graph is \(5\)-choosable. J. Comb. Theory Ser. B 62, 180–181 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Vizing, V.G.: Vertex colorings with given colors. Metody Diskret Anal. 29, 3–10 (1976). (in Russian)Google Scholar
  10. 10.
    Voigt, M.: List colourings of planar graphs. Discrete Math. 120, 215–219 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Xu, R., Wu, J.L.: A sufficient condition for a planar graph to be \(4\)-choosable. Discrete Appl. Math. 224, 120–122 (2017)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2019

Authors and Affiliations

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

Personalised recommendations