Planar Graphs Without 4-Cycles Adjacent to Triangles are DP-4-Colorable

  • Seog-Jin Kim
  • Xiaowei YuEmail author
Original Paper


DP-coloring (also known as correspondence coloring) of a simple graph is a generalization of list coloring. It is known that planar graphs without 4-cycles adjacent to triangles are 4-choosable, and planar graphs without 4-cycles are DP-4-colorable. In this paper, we show that planar graphs without 4-cycles adjacent to triangles are DP-4-colorable, which implies the two results above.


Coloring List-coloring DP-coloring Signed graph 



This paper was written as part of Konkuk University’s research support program for its faculty on sabbatical leave in 2018 (S.-J. Kim). The second author is supported by Science Foundation of Jiangsu Normal University (18XLRX020), the National Natural Science Foundation of China (11871311) and the Shandong Provincial Natural Science Foundation of China (ZR2018BA010, ZR2018MA001).


  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. Discret. Math. 339, 2680–2692 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bernshteyn, A., Kostochka, A.: On differences between DP-coloring and list coloring. arXiv:1705.04883v2
  4. 4.
    Bernshteyn, A., Kostochka, A., Pron, S.: On DP-coloring of graphs and multigraphs. Sib. Math. J. 58, 28–36 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bernshteyn, A., Kostochka, A., Zhu, X.: DP-colorings of graphs with high chromatic number. Eur. J. Comb. 65, 122–129 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cheng, P., Chen, M., Wang, Y.: Planar graphs without 4-cycles adjacent to triangles are 4-choosable. Discret. Math. 339(12), 3052–3057 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    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
  8. 8.
    Erdős, P., Rubin, A.L., Taylor, H.: Choosability in graphs. In: Proceedings of the West Coast Conference on Combinatorics, Graph Theory and Computing (Humboldt State University, Arcata, California, 1979), pp. 125–157, Congress. Numer., XXVI, Utilitas Mathematica Publishing Inc. Winnipeg, Manitoba (1980)Google Scholar
  9. 9.
    Jin, L., Kang, Y., Steffen, E.: Choosability in signed planar graphs. Eur. J. Comb. 52, 234–243 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Kim, S.-J., Ozeki, K.: A note on a Brooks’ type theorem for DP-coloring. J. Graph Theory (2018). Google Scholar
  11. 11.
    Kim, S.-J., Ozeki, K.: A Sufficient condition for DP-4-colorability. Discret. Math. 341(7), 1983–1986 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Lam, P.C.-B., Xu, B., Liu, J.: The 4-choosability of plane graphs without 4-cycles. J. Comb. Theory Ser. B 76(1), 117–126 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Máčajová, E., Raspaud, A., Škoviera, M.: The chromatic number of a signed graph. Electron. J. Comb. 23, #P1.14 (2016)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Thomassen, C.: Every planar graph is 5-choosable. J. Comb. Theory Ser. B 62(1), 180–181 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Vizing, V.G.: Vertex colorings with given colors (in Russian). Diskret. Analiz. 29, 3–10 (1976)Google Scholar
  16. 16.
    Voigt, M.: A not 3-choosable planar graph without 3-cycles. Discret. Math. 146, 325–328 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Zaslavsky, T.: Signed graph coloring. Discret. Math. 39, 215–228 (1982)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Japan KK, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematics EducationKonkuk UniversitySeoulKorea
  2. 2.School of Mathematics and StatisticsJiangsu Normal UniversityXuzhouPeople’s Republic of China

Personalised recommendations