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On (p, 1)-total labelling of 1-planar graphs

  • Xin ZhangEmail author
  • Yong Yu
  • Guizhen Liu
Research Article

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, it is proved that the (p, 1)-total labelling number of every 1-planar graph G is at most Δ(G) + 2p − 2 provided that Δ(G) ≥ 8p+4 or Δ(G) ≥ 6p+2 and g(G) ≥ 4. As a consequence, the well-known (p, 1)-total labelling conjecture has been confirmed for some 1-planar graphs.

Keywords

1-planar graph Alternating subgraph Master Total labelling Discharging 

MSC

05C10 05C15 

References

  1. [1]
    Albertson M.O., Mohar B., Coloring vertices and faces of locally planar graphs, Graphs Combin., 2006, 22(3), 289–295MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    Bazzaro F., Montassier M., Raspaud A., (d, 1)-total labelling of planar graphs with large girth and high maximum degree, Discrete Math., 2007, 307(16), 2141–2151MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    Bondy J.A., Murty U.S.R., Graph Theory with Applications, American Elsevier, New York, 1976Google Scholar
  4. [4]
    Borodin O.V., Solution of the Ringel problem on vertex-face coloring of planar graphs and coloring of 1-planar graphs, Metody Diskret. Analiz, 1984, 41, 12–26 (in Russian)MathSciNetzbMATHGoogle Scholar
  5. [5]
    Borodin O.V., On the total coloring of planar graphs, J. Reine Angew. Math., 1989, 394, 180–185MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    Borodin O.V., A new proof of the 6 color theorem, J. Graph Theory, 1995, 19(4), 507–521MathSciNetzbMATHCrossRefGoogle Scholar
  7. [7]
    Borodin O.V., Dmitriev I.G., Ivanova A.O., The height of a cycle of length 4 in 1-planar graphs with minimal degree 5 without triangles, Diskretn. Anal. Issled. Oper., 2008, 15(1), 11–16 (in Russian)MathSciNetGoogle Scholar
  8. [8]
    Borodin O.V., Kostochka A.V., Raspaud A., Sopena E., Acyclic coloring of 1-planar graphs, Discrete Appl. Math., 2001, 114(1–3), 29–41MathSciNetzbMATHCrossRefGoogle Scholar
  9. [9]
    Borodin O.V., Kostochka A.V., Woodall D.R., Total colorings of planar graphs with large maximum degree, J. Graph Theory, 1997, 26(1), 53–59MathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    Borodin O.V., Kostochka A.V., Woodall D.R., List edge and list total colourings of multigraphs, J. Combin. Theory Ser. B, 1997, 71(2), 184–204MathSciNetCrossRefGoogle Scholar
  11. [11]
    Borodin O.V., Kostochka A.V., Woodall D.R., Total colourings of planar graphs with large girth, European. J. Combin., 1998, 19(1), 19–24MathSciNetzbMATHCrossRefGoogle Scholar
  12. [12]
    Calamoneri T., The L(h, k)-labelling problem: a survey and annotated bibliography, Comput. J., 2006, 49(5), 585–608CrossRefGoogle Scholar
  13. [13]
    Chen D., Wang W., (2, 1)-total labelling of outerplanar graphs, Discrete Appl. Math., 2007, 155(18), 2585–2593MathSciNetzbMATHCrossRefGoogle Scholar
  14. [14]
    Fabrici I., Madaras T., The structure of 1-planar graphs, Discrete Math., 2007, 307(7–8), 854–865MathSciNetzbMATHCrossRefGoogle Scholar
  15. [15]
    Hasunuma T., Ishii T., Ono H., Uno Y., The (2,1)-total labeling number of outerplanar graphs is at most Δ + 2, In: Combinatorial Algorithms, Lecture Notes in Comput. Sci., 6460, Springer, Berlin, 2011, 103–106CrossRefGoogle Scholar
  16. [16]
    Havet F., (d, 1)-total labelling of graphs, In: Workshop on Graphs and Algorithms, Dijon, 2003Google Scholar
  17. [17]
    Havet F., Yu M.-L., (d, 1)-total labelling of graphs, INRIA, 2002, Technical Report #4650Google Scholar
  18. [18]
    Havet F., Yu M.-L., (p, 1)-total labelling of graphs, Discrete Math., 2008, 308(4), 496–513MathSciNetzbMATHCrossRefGoogle Scholar
  19. [19]
    Hudák D., Madaras T., On local structures of 1-planar graphs of minimum degree 5 and girth 4, Discuss. Math. Graph Theory, 2009, 29(2), 385–400MathSciNetzbMATHGoogle Scholar
  20. [20]
    Hudák D., Madaras T., On local properties of 1-planar graphs with high minimum degree, Ars Math. Contemp., 2011, 4(2), 245–254Google Scholar
  21. [21]
    Kowalik Ł., Sereni J.S., Škrekovski R., Total-coloring of plane graphs with maximum degree nine, SIAM J. Discrete Math., 2008, 22(4), 1462–1479MathSciNetzbMATHCrossRefGoogle Scholar
  22. [22]
    Montassier M., Raspaud A., (d; 1)-total labeling of graphs with a given maximum average degree, J. Graph Theory, 2006, 51(2), 93–109MathSciNetzbMATHCrossRefGoogle Scholar
  23. [23]
    Ringel G., Ein Sechsfarbenproblem auf der Kugel, Abh. Math. Semin. Univ. Hamburg, 1965, 29, 107–117MathSciNetzbMATHCrossRefGoogle Scholar
  24. [24]
    Sanders D. P., Zhao Y., On total 9-coloring planar graphs of maximum degree seven, J. Graph Theory, 1999, 31(1), 67–73MathSciNetzbMATHCrossRefGoogle Scholar
  25. [25]
    Wang W., Lih K.-W., Coupled choosability of plane graphs, J. Graph Theory, 2008, 58(1), 27–44MathSciNetzbMATHCrossRefGoogle Scholar
  26. [26]
    Whittlesey M.A., Georges J.P., Mauro D.W., On the λ-number of Q n and related graphs, SIAM J. Discrete Math, 1995, 8(4), 499–506MathSciNetzbMATHCrossRefGoogle Scholar
  27. [27]
    Wu J., Wang P., List-edge and list-total colorings of graphs embedded on hyperbolic surfaces, Discrete Math., 2008, 308(24), 6210–6215MathSciNetzbMATHCrossRefGoogle Scholar
  28. [28]
    Yeh R.K., A survey on labeling graphs with a condition at distance two, Discrete Math., 2006, 306(12), 1217–1231MathSciNetzbMATHCrossRefGoogle Scholar
  29. [29]
    Zhang X., Liu G., On edge colorings of 1-planar graphs without chordal 5-cycles, Ars Combin. (in press)Google Scholar
  30. [30]
    Zhang X., Liu G., On edge colorings of 1-planar graphs without adjacent triangles, preprint available at http://xinzhang.hpage.com/get_file.php?id=1283123&vnr=529770
  31. [31]
    Zhang X., Liu G, Wu J.-L., Edge coloring of triangle-free 1-planar graphs, J. Shandong Univ. Nat. Sci., 2010, 45(6), 15–17 (in Chinese)MathSciNetzbMATHGoogle Scholar
  32. [32]
    Zhang X., Liu G., Wu J.-L., Structural properties of 1-planar graphs and an application to acyclic edge coloring, Scientia Sinica Mathematica, 2010, 40(10), 1025–1032 (in Chinese)Google Scholar
  33. [33]
    Zhang X., Liu G., Wu J.-L., Light subgraphs in the family of 1-planar graphs with high minimum degree, Acta Math. Sin. (Engl. Ser.) (in press)Google Scholar
  34. [34]
    Zhang X., Wu J.-L., On edge colorings of 1-planar graphs, Inform. Process. Lett., 2011, 111(3), 124–128MathSciNetCrossRefGoogle Scholar
  35. [35]
    Zhang X., Wu J.-L., Liu G., List edge and list total coloring of 1-planar graphs, preprint available at http://xinzhang.hpage.com/get_file.php?id=1251999&vnr=768295

Copyright information

© © Versita Warsaw and Springer-Verlag Wien 2011

Authors and Affiliations

  1. 1.School of MathematicsShandong UniversityJinanChina

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