Advertisement

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

  • Pongpat Sittitrai
  • Kittikorn NakprasitEmail author
Original Paper
  • 20 Downloads

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.

Keywords

DP-coloring Improper colorings Planar graphs 

Mathematics Subject Classification

05C15 

Notes

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.

References

  1. 1.
    Bernshteyn, A., Kostochka, A., Pron, S.: On DP-coloring of graphs and multigraphs. Sib. Math. J. 58(1), 28–36 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Chen, M., Raspaud, A.: On (3, 1)*-choosability of planar graphs without adjacent short cycles. Discrete Appl. Math. 162, 159–166 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Chen, M., Raspaud, A., Wang, W.: A (3, 1)*-choosable theorem on planar graphs. J. Comput. Optim. 32(3), 927–940 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Dong, W., Xu, B.: A note on list improper coloring of plane graphs. Discrete Appl. Math. 157(2), 433–436 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Eaton, E., Hull, T.: Defective list colorings of planar graphs. Bull. Inst. Combin. Appl. 25, 79–87 (1999)MathSciNetzbMATHGoogle Scholar
  8. 8.
    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)Google Scholar
  9. 9.
    Kim, S.-J., Ozeki, K.: A sufficient condition for DP-\(4\)-colorability. Discrete Math. 341(7), 1983–1986 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Kim, S.-J., Yu, X.: Planar graphs without \(4\)-cycles adjacent to triangles are DP-\(4\)-colorable (2017). arXiv:1712.08999
  11. 11.
    Lih, K., Song, Z., Wang, W., Zhang, K.: A note on list improper coloring planar graphs. Appl. Math. Lett. 14(3), 269–273 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Škrekovski, R.: List improper colourings of planar graphs. Combin. Probab. Comput. 8(3), 293–299 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Thomassen, C.: Every planar graph is \(5\)-choosable. J. Combin. Theory Ser. B 62(1), 180–181 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Vizing, V.G.: Vertex colorings with given colors. Metody Diskret. Analiz. 29, 3–10 (1976)Google Scholar
  15. 15.
    Voigt, M.: List colourings of planar graphs. Discrete Math. 120(1–3), 215–219 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Wang, Y., Xu, L.: Improper choosability of planar graphs without 4-cycles. SIAM J. Discrete Math. 27(4), 2029–2037 (2013)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Japan KK, part of Springer Nature 2019

Authors and Affiliations

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

Personalised recommendations