Skip to main content
SpringerLink
Log in

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

  • Original Paper
  • Published:
Graphs and Combinatorics Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  1. Alon, N.: Degrees and choice numbers. Random Struct. Algorithms 16, 364–368 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bernshteyn, A.: The asymptotic behavior of the correspondence chromatic number. Discret. Math. 339, 2680–2692 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bernshteyn, A., Kostochka, A.: On differences between DP-coloring and list coloring. arXiv:1705.04883v2

  4. Bernshteyn, A., Kostochka, A., Pron, S.: On DP-coloring of graphs and multigraphs. Sib. Math. J. 58, 28–36 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bernshteyn, A., Kostochka, A., Zhu, X.: DP-colorings of graphs with high chromatic number. Eur. J. Comb. 65, 122–129 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  9. Jin, L., Kang, Y., Steffen, E.: Choosability in signed planar graphs. Eur. J. Comb. 52, 234–243 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  10. Kim, S.-J., Ozeki, K.: A note on a Brooks’ type theorem for DP-coloring. J. Graph Theory (2018). https://doi.org/10.1002/jgt.22425

    Google Scholar 

  11. Kim, S.-J., Ozeki, K.: A Sufficient condition for DP-4-colorability. Discret. Math. 341(7), 1983–1986 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  13. Máčajová, E., Raspaud, A., Škoviera, M.: The chromatic number of a signed graph. Electron. J. Comb. 23, #P1.14 (2016)

    MathSciNet  MATH  Google Scholar 

  14. Thomassen, C.: Every planar graph is 5-choosable. J. Comb. Theory Ser. B 62(1), 180–181 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  15. Vizing, V.G.: Vertex colorings with given colors (in Russian). Diskret. Analiz. 29, 3–10 (1976)

    Google Scholar 

  16. Voigt, M.: A not 3-choosable planar graph without 3-cycles. Discret. Math. 146, 325–328 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  17. Zaslavsky, T.: Signed graph coloring. Discret. Math. 39, 215–228 (1982)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

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).

Author information

Authors and Affiliations

  1. Department of Mathematics Education, Konkuk University, Seoul, 05029, Korea

    Seog-Jin Kim

  2. School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, 221116, Jiangsu, People’s Republic of China

    Xiaowei Yu

Authors
  1. Seog-Jin Kim
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiaowei Yu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xiaowei Yu.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kim, SJ., Yu, X. Planar Graphs Without 4-Cycles Adjacent to Triangles are DP-4-Colorable. Graphs and Combinatorics 35, 707–718 (2019). https://doi.org/10.1007/s00373-019-02028-z

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00373-019-02028-z

Keywords

Search

Navigation