Skip to main content
SpringerLink
Log in

A totally (Δ + 1)-colorable 1-planar graph with girth at least five

  • Published:
Acta Mathematica Sinica, English Series Aims and scope Submit manuscript

Abstract

A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. In this paper, we prove that every 1-planar graph G with maximum degree Δ(G) ≥ 12 and girth at least five is totally (Δ(G) + 1)-colorable.

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

Similar content being viewed by others

References

  1. Behzad, M.: Graphs and Their Chromatic Numbers, Doctoral thesis, Michigan State University, 1965

    Google Scholar 

  2. Bondy, J. A., Murty, U. S. R.: Graph Theory with Applications, North-Holland, New York, 1976

    Book  MATH  Google Scholar 

  3. Borodin, O. V.: On the total coloring of planar graphs. J. Reine Angew. Math., 394, 180–185 (1989)

    MathSciNet  MATH  Google Scholar 

  4. Borodin, O. V., Kostochka, A. V., Woodall, D. R.: List edge and list total colorings of multigraphs. J. Combin. Theory Ser. B, 71, 184–204 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  5. Borodin, O. V., Kostochka, A. V., Woodall, D. R.: Total colorings of planar graphs with large maximum degree. J. Graph Theory, 26, 53–59 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  6. Czap, J.: A note on total colorings of 1-planar graphs. Inform. Process. Lett., 113, 516–517 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ding, L. H., Wang, G. H., Yan, G. Y.: Neighbor sum distinguishing total colorings via the Combinatorial Nullstellensatz. Sci. China Ser. A, 57 9, 1875–1882 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  8. Hou, J. F., Zhu, Y., Liu, G. Z., et al.: Total colorings of planar graphs without small cycles. Graphs Combin., 24, 91–100 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  9. Kostochka, A. V.: The total chromatic number of any multigraph with maximum degree five is at most seven. Discrete Math., 162, 199–214 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  10. Kowalik, L., Sereni, J.-S., Škrekovski R.: Total-coloring of plane graphs with maximum degree nine. SIAM J. Discrete Math., 22, 1462–1479 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Li, H. L., Ding, L. H, Liu, B. Q., et al.: Neighbor sum distinguishing total colorings of planar graphs. J. Comb. Optim., 60 4, 777–791 (2014)

    MathSciNet  Google Scholar 

  12. Ringel, G.: Ein Sechsfarbenproblem auf der Kugel. Abh. Math. Sem. Univ. Hamburg, 29, 107–117 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  13. Rosenfeld, M.: On the total coloring of certain graphs. Israel J. Math., 9, 396–402 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  14. Shen, L., Wang, Y.: Planar graphs with maximum degree 7 and without 5-cycles are 8-totally-colorable. Discrete Math., 310(17–18), 2372–2379 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  15. Sanchez-Arroyo, A.: Determining the total coloring number is NP-hard. Discrete Math., 78, 315–319 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  16. Vijayaditya, N.: On total chromatic number of a graph. J. London Math. Soc., 3 2, 405–408 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  17. Vizing, V. G.: Some unresolved problems in graph theory (in Russian). Uspekhi Mat. Nauk, 23, 117–134 (1968)

    MathSciNet  MATH  Google Scholar 

  18. Wang, B., Wu, J. L., Wang, H. J.: Total colorings of planar graphs without chordal 6-cycles. Discrete Appl. Math., 171, 116–121 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  19. Wang, W. F.: Total chromatic number of planar graphs with maximum degree ten. J. Graph Theory, 54, 91–102 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  20. Wang, H. J., Wu, L. D., Wu, J. L.: Minimum total coloring of planar graph. J. Global Optim., 30 1, 160–173 (2015)

    Google Scholar 

  21. Zhang, X., Wu, J. L.: On edge colorings of 1-planar graphs. Inform. Process. Lett., 111, 124–128 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  22. Zhang, X., Wu, J. L., Liu, Z. G.: List edge and list total coloring of 1-planar graphs. Front. Math. China, 7 5, 1005–1018 (2012)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics, Shandong University, Ji’nan, 250100, P. R. China

    Lin Sun & Jian Liang Wu

  2. Department of Mathematics, Changji University, Changji, 831100, P. R. China

    Lin Sun & Hua Cai

Authors
  1. Lin Sun
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jian Liang Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hua Cai
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jian Liang Wu.

Additional information

The second author is supported by National Natural Science Foundation of China (Grant No. 11271006)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sun, L., Wu, J.L. & Cai, H. A totally (Δ + 1)-colorable 1-planar graph with girth at least five. Acta. Math. Sin.-English Ser. 32, 1337–1349 (2016). https://doi.org/10.1007/s10114-016-5480-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10114-016-5480-9

Keywords

MR(2010) Subject Classification

Search

Navigation