Advertisement

Acta Mathematica Sinica, English Series

, Volume 29, Issue 7, pp 1421–1428 | Cite as

On edge colorings of 1-toroidal graphs

  • Xin ZhangEmail author
  • Gui Zhen Liu
Article
  • 126 Downloads

Abstract

A graph is 1-toroidal, if it can be embedded in the torus so that each edge is crossed by at most one other edge. In this paper, it is proved that every 1-toroidal graph with maximum degree Δ ≥ 10 is of class one in terms of edge coloring. Meanwhile, we show that there exist class two 1-toroidal graphs with maximum degree Δ for each Δ ≤8.

Keywords

1-Toroidal graph 1-planar graph edge coloring 

MR(2010) Subject Classification

05C10 05C15 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Bondy, J. A., Murty, U. S. R.: Graph Theory with Applications, North-Holland, New York, 1976zbMATHGoogle Scholar
  2. [2]
    Sanders, D. P., Zhao, Y.: Planar graphs of maximum degree seven are class I. J. Combin. Theory Ser. B, 83, 201–212 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    Zhang, L.: Every planar graph with maximum degree 7 is of class 1. Graphs Combin., 16, 467–495 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  4. [4]
    Vizing, V. G.: Critical graphs with given chromatic class. Diskret. Analiz., 5, 9–17 (1965)MathSciNetzbMATHGoogle Scholar
  5. [5]
    Ringel, G.: Ein Sechsfarbenproblem auf der Kugel (in German). Abh. Math. Semin. Univ. Hambg., 29, 107–117 (1965)MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    Borodin, O. V.: Solution of Ringel’s problems on the vertex-face coloring of plane graphs and on the coloring of 1-planar graphs. Diskret. Analiz., 41, 12–26 (1984)MathSciNetzbMATHGoogle Scholar
  7. [7]
    Borodin, O. V.: A new proof of the 6-color theorem. J. Graph Theory, 19(4), 507–521 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  8. [8]
    Borodin, O. V., Dmitriev, I. G., Ivanova, A. O.: The height of a cycle of length 4 in 1-planar graphs with minimum degree 5 without triangles (in Russian). Diskretn. Anal. Issled. Oper., 15(1), 11–16 (2008)MathSciNetzbMATHGoogle Scholar
  9. [9]
    Borodin, O. V., Kostochka, A. V., Raspaud, A., et al.: Acyclic colouring of 1-planar graphs. Discrete Appl. Math., 114, 29–41 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    Chen, Z.-Z., Kouno, M.: A linear-time algorithm for 7-coloring 1-plane graphs. Algorithmica, 43(3), 147–177 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  11. [11]
    Fabrici, I., Madaras, T.: The structure of 1-planar graphs. Discrete Math., 307, 854–865 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  12. [12]
    Hudák, D., Madaras, T.: On local structures of 1-planar graphs of minimum degree 5 and girth 4. Discuss. Math. Graph Theory, 29, 385–400 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  13. [13]
    Hudák, D., Madaras, T.: On local properties of 1-planar graphs with high minimum degree. Ars Math. Contemp., 4(2), 245–254 (2011)MathSciNetzbMATHGoogle Scholar
  14. [14]
    Hudák, D., Šugerek P.: Light edges in 1-planar graphs with prescribed minimum degree. Discuss. Math. Graph Theory, 32(3), 545–556 (2012)MathSciNetCrossRefGoogle Scholar
  15. [15]
    Zhang, X., Liu, G.: On edge colorings of 1-planar graphs without adjacent triangles. Inform. Process. Lett., 112, 138–142 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  16. [16]
    Zhang, X., Liu, G.: On edge colorings of 1-planar graphs without chordal 5-cycles. Ars Combin., 104, 431–436 (2012)MathSciNetzbMATHGoogle Scholar
  17. [17]
    Zhang, X., Liu, G., Wu, J.-L.: Edge coloring of triangle-free 1-planar graphs (in Chinese). J. Shandong Univ. Nat. Sci., 45(6), 15–17 (2010)MathSciNetzbMATHGoogle Scholar
  18. [18]
    Zhang, X., Liu, G., Wu, J.-L.: (1, λ)-embedded graphs and the acyclic edge choosability. Bull. Korean Math. Soc., 49(3), 573–580 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  19. [19]
    Zhang, X., Wu, J.-L.: On edge colorings of 1-planar graphs. Inform. Process. Lett., 111(3), 124–128 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  20. [20]
    Zhang, X., Liu, G., Wu, J.-L.: On the linear arboricity of 1-planar graphs. OR Trans., 15(3), 38–44 (2011)MathSciNetzbMATHGoogle Scholar
  21. [21]
    Zhang, X., Liu, G., Wu, J.-L.: Light subgraphs in the family of 1-planar graphs with high minimum degree. Acta Mathematica Sinica, English series, 28(6), 1155–1168 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  22. [22]
    Zhang, X., Wu, J.-L., Liu, G.: New upper bounds for the heights of some light subgraphs in 1-planar graphs with high minimum degree. Discrete Math. Theor. Comput. Sci., 13(3), 9–16 (2011)MathSciNetGoogle Scholar
  23. [23]
    Zhang, X., Wu, J.-L., Liu, G.: List edge and list total coloring of 1-planar graphs. Front. Math. China, 7(5), 1005–1018 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  24. [24]
    Zhang, X., Yu, Y., Liu, G.: On (p, 1)-total labelling of 1-planar graphs. Cent. Eur. J. Math., 9(6), 1424–1434 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  25. [25]
    Luo, R., Miao, L., Zhao, Y.: The size of edge chromatic critical graphs with maximum degree 6. J. Graph Theory, 60, 149–171 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  26. [26]
    Vizing, V. G.: Some unsolved problems in graph theory (in Russian). Uspekhi Mat. Nauk., 23, 117–134 (1968); English translation in Russian Math. Surveys, 23, 125–141 (1968)MathSciNetzbMATHGoogle Scholar

Copyright information

© Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of MathematicsXidian UniversityXi’anP. R. China
  2. 2.School of MathematicsShandong UniversityJi’nanP. R. China

Personalised recommendations