A note on edge-choosability of planar graphs without intersecting 4-cycles

  • Qiaoling Ma
  • Jihui Wang
  • Jiansheng Cai
  • Sumei Zhang
Article
  • 56 Downloads

Abstract

A graph G is edge-L-colorable, if for a given edge assignment L={L(e):eE(G)}, there exits a proper edge-coloring φ of G such that φ(e)∈L(e) for all eE(G). If G is edge-L-colorable for every edge assignment L with |L(e)|≥k for eE(G), then G is said to be edge-k-choosable. In this paper, We investigate structural of planar graphs without intersecting 4-cycles and show that every planar graph without intersecting 4-cycles is edge-k-choosable, where \(k=\max\{7,\Delta(G)+1\}\).

Keywords

Planar graph Edge-coloring Choosability Cycle Chord Combinatorial problem 

Mathematics Subject Classification (2000)

05C15 

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References

  1. 1.
    Häggkvist, R., Chetwynd, A.: Some upper bounds on the total and list chromatic numbers of multigraphs. J. Graph Theory 16, 503–516 (1992) MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Bollobás, B., Harris, A.J.: List-colourings of graphs. Graph Comb. 1, 115–127 (1985) MATHCrossRefGoogle Scholar
  3. 3.
    Galvin, F.: The list chromatic index of a bipartite multigraph. J. Comb. Theory, Ser. B 63, 153–158 (1995) MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Häggkvist, R., Janssen, J.: New bounds on the list-chromatic index of the complete graph and other simple graphs. Comb. Probab. Comput. 6, 295–313 (1997) MATHCrossRefGoogle Scholar
  5. 5.
    Borodin, O.V., Kostochka, A.V., Woodall, D.R.: List edge and list total colourings of multigraphs. J. Comb. Theory, Ser. B 71, 184–204 (1997) MathSciNetCrossRefGoogle Scholar
  6. 6.
    Wang, W.F., Lih, K.W.: Choosability, edge choosability and total choosability of outerplanar graphs. Eur. J. Comb. 22, 71–78 (2001) MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Wu, B.R., An, X.H.: A note on edge choosability and degeneracy of planar graphs. Lect. Notes Comput. Sci. 5573, 249–257 (2009) CrossRefGoogle Scholar
  8. 8.
    Vizing, V.G.: On an estimate of the chromatic class of p-graph. Metod. Diskretn. Anal. 3, 25–30 (1964) (in Russian) MathSciNetGoogle Scholar
  9. 9.
    Harris, A.J.: Problems and conjectures in extremal graph theory. Ph.D. Dissertation, Cambridge University, UK, 1984 Google Scholar
  10. 10.
    Juvan, M., Mohar, B., Škrekovski, R.: Graphs of degree 4 are 5-choosable. J. Graph Theory 32, 250–262 (1999) MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Kostochka, A.V.: List edge chromatic number of graphs with large girth. Discrete Math. 101, 189–201 (1992) MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Borodin, O.V.: An extension of Kotzig’s’ theorem and the list edge coloring of planar graphs. Mat. Zametki 48, 22–48 (1990) (in Russian) MathSciNetGoogle Scholar
  13. 13.
    Shen, Y., Zheng, G., He, W., Zhao, Y.: Structural properties and edge choosability of planar graphs without 4-cycles. Discrete Math. 308, 5789–5794 (2008) MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Wang, W.F., Lih, K.W.: Structural properties and edge choosability of planar graphs without 6-cycles. Comb. Probab. Comput. 10, 267–276 (2001) MathSciNetMATHGoogle Scholar
  15. 15.
    Wang, W.F., Lih, K.W.: Choosability and edge choosability of planar graphs without intersecting triangles. SIAM J. Discrete Math. 15, 538–545 (2002) MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Wang, W.F., Lih, K.W.: Choosability and edge choosability of planar graphs without five cycles. Appl. Math. Lett. 15, 561–565 (2002) MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Zhang, L., Baoyindureng: Edge choosability of planar graphs without small cycles. Discrete Math. 283, 289–293 (2004) MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Hou, J.F., Liu, G.Z., Cai, J.S.: List Edge and List Total colorings of planar graphs without 4-cycles. Theor. Comput. Sci. 369, 250–255 (2006) MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Liu, B., Hou, J.F., Liu, G.Z.: List Edge and List Total colorings of planar graphs without short cycles. Inf. Process. Lett. 108, 347–351 (2008) MathSciNetCrossRefGoogle Scholar
  20. 20.
    Hou, J.F., Liu, G.Z., Cai, J.S.: Edge-choosability of planar graphs without adjacent triangles or 7-cycles. Discrete Math. 309, 77–84 (2009) MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Korean Society for Computational and Applied Mathematics 2010

Authors and Affiliations

  • Qiaoling Ma
    • 1
  • Jihui Wang
    • 1
  • Jiansheng Cai
    • 2
  • Sumei Zhang
    • 1
  1. 1.School of ScienceUniversity of JinanJinanP.R. China
  2. 2.School of Mathematics and Information ScienceWeifang UniversityWeifangP.R. China

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