Equitable Coloring of Three Classes of 1-planar Graphs

  • Xin ZhangEmail author
  • Hui-juan Wang
  • Lan Xu


A graph is 1-planar if it can be drawn on a plane so that each edge is crossed by at most one other edge. A plane graph with near-independent crossings or independent crossings, say NIC-planar graph or IC-planar graph, is a 1-planar graph with the restriction that for any two crossings the four crossed edges are incident with at most one common vertex or no common vertices, respectively. In this paper, we prove that each 1-planar graph, NIC-planar graph or IC-planar graph with maximum degree Δ at least 15, 13 or 12 has an equitable Δ-coloring, respectively. This verifies the well-known Chen-Lih-Wu Conjecture for three classes of 1-planar graphs and improves some known results.


1-planar graph equitable coloring independent crossing 

2000 MR Subject Classification

05C15 05C10 


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  1. [1]
    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) (in Russian)zbMATHGoogle Scholar
  2. [2]
    Borodin, O.V. A new proof of the six color theorem. Journal of Graph Theory, 19(4): 507–521 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    Cazap, J., Šugerek, P. Drawing Graph Joins in the Plane with Restrictions on Crossings. Filomat, 31: 363–370 (2017)MathSciNetCrossRefGoogle Scholar
  4. [4]
    Chen, B.L., Lih, K.W., Wu, P.L. Equitable coloring and the maximum degree. European J. Combin., 15: 443–447 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    Chen, B.L., Lih, K.W., Yan, J.H. A note on equitable coloring of interval graphs. Manuscript, 1998Google Scholar
  6. [6]
    Chen, B.L., Yen, C.H. Equitable Δ-coloring of graphs. Discrete Math., 312: 1512–1517 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    Fabrici, I., Madaras, T. The structure of 1-planar graphs. Discrete Math., 307: 854–865 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    Hajnal, A., Szemerédi, E. Proof of a conjecture of P. Erdős. In: Combinatorial Theory and its Applications, P. Erdős, A. Rényi and V. T. Sós, eds, North-Holand, London, 1970, 601–623Google Scholar
  9. [9]
    Kierstead, H.A, Kostochka, A.V. Every 4-colorable graph with maximum degree 4 has an equitable 4-coloring. J. Graph Theory, 71: 31–48 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    Kierstead, H.A., Kostochka, A.V. A refinement of a result of Corrádi and Hajnal. Combinatorica, 35(4): 497–512 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    Kostochka, A.V., Nakprasit, K. On equitable Δ-coloring of graphs with low average degree. Theoret. Comput. Sci., 349: 82–91 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    Kostochka, A.V., Nakprasit, K. Equitable colorings of k-degenerate graphs. Combin. Probab. Comput., 12: 53–60 (2013)zbMATHGoogle Scholar
  13. [13]
    Král, D., Stacho, L. Coloring plane graphs with independent crossings. J. Graph Theory, 64(3): 184–205 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    Lih, K.W. Equitable coloring of graphs. In: Handbook of Combinatorial Optimization (P. M. Pardalos, D.-Z. Du, R. Graham, eds), 2nd ed., Springer, 2013, 1199–1248CrossRefGoogle Scholar
  15. [15]
    Lih, K.W., Wu, P.L. On equitable coloring of bipartite graphs. Discrete Math., 151: 155–160 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    Nakprasit, K. Equitable colorings of planar graphs with maximum degree at least nine. Discrete Math., 312: 1019–1024 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    Ringel, G. Ein sechsfarbenproblem auf der Kugel. Abh. Math. Sem. Hamburg. Univ., 29: 107–117 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    Tian, J., Zhang, X. Pseudo-outerplanar graphs and chromatic conjectures. Ars Combin., 114: 353–361 (2014)MathSciNetzbMATHGoogle Scholar
  19. [19]
    Yap, H.P., Zhang, Y. The equitable Δ-coloring coniecture holds for outerplanar graphs. Bull. Inst. Math. Acad. Sin.(N.S.), 25: 143–149 (1997)Google Scholar
  20. [20]
    Yap, H.P., Zhang, Y. Equitable colorings of planar graphs. J. Combin. Math. Combin. Comput., 27: 97–105 (1998)MathSciNetzbMATHGoogle Scholar
  21. [21]
    Zhang, X. Drawing complete multipartite graphs on the plane with restrictions on crossings. Acta Math. Sin. (Engl. Ser.), 30(12): 2045–2053 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    Zhang, X. On equitable colorings of sparse graphs. Bull. Malays. Math. Sci. Soc., 39(1): 257–268 (2016)MathSciNetCrossRefGoogle Scholar
  23. [23]
    Zhang, X., Liu, G. The structure of plane graphs with independent crossings and its appications to coloring problems. Cent. Eur. J. Math., 11(2): 308–321 (2013)MathSciNetzbMATHGoogle Scholar
  24. [24]
    Zhang, X., Wu, J.L, On equitable and equitable list colorings of series-parallel graphs. Discrete Math., 311: 800–803 (2011)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Institute of Applied Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsXidian UniversityXi’anChina
  2. 2.College of MathematicsQingdao UniversityQingdaoChina
  3. 3.Department of MathematicsChangji UniversityChangjiChina

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