Journal of Combinatorial Optimization

, Volume 35, Issue 2, pp 555–562 | Cite as

List-edge-coloring of planar graphs without 6-cycles with three chords



A graph G is edge-k-choosable if, whenever we are given a list L(e) of colors with \(|L(e)|\ge k\) for each \(e\in E(G)\), we can choose a color from L(e) for each edge e such that no two adjacent edges receive the same color. In this paper we prove that if G is a planar graph, and each 6-cycle contains at most two chords, then G is edge-k-choosable, where \(k=\max \{8,\Delta (G)+1\}\), and edge-t-choosable, where \(t=\max \{10,\Delta (G)\}\).


Edge-choosable List-edge-chromatic-number Cycle Planar graph 



We would like to thank the referees for providing some very helpful comments and suggestions for revising this paper.


  1. Bonamy M (2015) Planar graphs with \(\Delta \ge 8\) are \((\Delta +1)\)-edge-choosable. SIAM J Discrete Math 29:1735–1763MathSciNetCrossRefMATHGoogle Scholar
  2. Bondy JA, Murty USR (1976) Graph theory with applications. Elsevier, New YorkGoogle Scholar
  3. Borodin OV, Kostochka AV, Woodall DR (1997) List edge and list total colourings of multigraphs. J Comb Theory Ser B 71:184–204MathSciNetCrossRefMATHGoogle Scholar
  4. Cai JS, Hou JF, Zhang X, Liu GZ (2009) Edge-choosability of planar graphs without non-induced \(5\)-cycles. Inf Process Lett 109:343–346MathSciNetCrossRefMATHGoogle Scholar
  5. Cai JS (2015) List edge coloring of planar graphs without non-induced \(6\)-cycles. Gr Comb 31:827–832MathSciNetCrossRefMATHGoogle Scholar
  6. Chang GJ, Hou J, Roussel N (2010) On the total choosability of planar graphs and of sparse graphs. Inf Process Lett 110:849–853MathSciNetCrossRefMATHGoogle Scholar
  7. Erdős P, Rubin AL, Taylor H (1979) Choosability in graphs. Congr Numer 26:125–157MATHGoogle Scholar
  8. Galvin F (1995) The list chromatic index of a bipartite multigraph. J Comb Theory Ser B 63:153–158MathSciNetCrossRefMATHGoogle Scholar
  9. Häggkvist R, Janssen J (1997) New bounds on the list-chromatic index of the complete graph and other simple graphs. Comb Probab Comput 6:295–313MathSciNetCrossRefMATHGoogle Scholar
  10. Häggkvist R, Chetwynd A (1992) Some upper bounds on the total and list chromatic numbers of multigraphs. J Gr Theory 16:503–516MathSciNetCrossRefMATHGoogle Scholar
  11. Harris AJ (1984) Problems and conjectures in extremal graph theory, Ph.D. Dissertation, Cambridge University, UKGoogle Scholar
  12. Hou JF, Liu GZ, Cai JS (2006) List edge and list total colorings of planar graphs without \(4\)-cycles. Theor Comput Sci 369:250–255MathSciNetCrossRefMATHGoogle Scholar
  13. Hou JF, Zhu Y, Liu GZ, Wu JL (2008) Total colorings of planar graphs without small cycles. Gr Comb 24:91–100MathSciNetCrossRefMATHGoogle Scholar
  14. Hou JF, Liu GZ, Cai JS (2009) Edge-choosability of planar graphs without adjacent triangles or without \(7\)-cycles. Discrete Math 309:77–84MathSciNetCrossRefMATHGoogle Scholar
  15. Jensen TR, Toft B (1995) Graph coloring problems. Wiley, New YorkMATHGoogle Scholar
  16. Juvan M, Mohar B, S̆rekovski R (1999) Graphs of degree \(4\) are \(5\)-choosable. J Graph Theory 32:250–262MathSciNetCrossRefMATHGoogle Scholar
  17. Kostochka AV (1992) List edge chromatic number of graphs with large girth. Discrete Math 101:189–201MathSciNetCrossRefMATHGoogle Scholar
  18. Liu B, Hou JF, Liu GZ (2008) List edge and list total colorings of planar graphs without short cycles. Inf Process Lett 108:347–351MathSciNetCrossRefMATHGoogle Scholar
  19. Shen Y, Zheng G, He W, Zhao Y (2008) Structural properties and edge choosability of planar graphs without \(4\)-cycles. Discrete Math 308:5789–5794MathSciNetCrossRefMATHGoogle Scholar
  20. Wang WF, Lih KW (2001) Structural properties and edge choosability of planar graphs without \(6\)-cycles. Comb Probab Comput 10:267–276MathSciNetMATHGoogle Scholar
  21. Wang WF, Lih KW (2001) Choosability, edge choosability and total choosability of outerplanar graphs. Eur J Comb 22:71–78CrossRefMATHGoogle Scholar
  22. Wang WF, Lih KW (2002) Choosability and edge choosability of planar graphs without five cycles. Appl Math Lett 15:561–565MathSciNetCrossRefMATHGoogle Scholar
  23. Woodall DR (1999) Edge-choosability of multicircuits. Discrete Math 202:271–277MathSciNetCrossRefMATHGoogle Scholar
  24. Wu JL, Wang P (2008) List-edge and list-total colorings of graphs embedded on hyperbolic surfaces. Discrete Math 308:6210–6215MathSciNetCrossRefMATHGoogle Scholar
  25. Zhang L, Wu B (2004) Edge choosability of planar graphs without small cycles. Discrete Math 283:289–293MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.School of ScienceChina University of Geosciences (Beijing)BeijingChina
  2. 2.School of MathematicsShandong UniversityJinanChina

Personalised recommendations