Journal of Combinatorial Optimization

, Volume 31, Issue 3, pp 1221–1240 | Cite as

The entire choosability of plane graphs

  • Weifan WangEmail author
  • Tingting Wu
  • Xiaoxue Hu
  • Yiqiao Wang


A plane graph \(G\) is entirely \(k\)-choosable if, for every list \(L\) of colors satisfying \(L(x)=k\) for all \(x\in V(G)\cup E(G) \cup F(G)\), there exists a coloring which assigns to each vertex, each edge and each face a color from its list so that any adjacent or incident elements receive different colors. In 1993, Borodin proved that every plane graph \(G\) with maximum degree \(\Delta \ge 12\) is entirely \((\Delta +2)\)-choosable. In this paper, we improve this result by replacing 12 by 10.


Plane graph Entire choosability Maximum degree 



The authors would like to thank the referees for their valuable comments that helped to improve this work. Weifan Wang: Research supported by NSFC (No. 11371328). Yiqiao Wang: Research supported by NSFC (No. 11301035)


  1. Borodin OV (1987) Coupled colorings of graphs on a plane. Metod Diskret Anal 45:21–27 (in Russian)MathSciNetzbMATHGoogle Scholar
  2. Borodin OV (1993) Simultaneous coloring of edge neighborhoods in planar graphs and the simultaneous coloring of vertices, edges and faces. Mat Zamet 53:35–47 (in Russian)zbMATHGoogle Scholar
  3. Borodin OV (1994) Simultaneous coloring of edges and faces of plane graphs. Discret Math 128:21–33MathSciNetCrossRefzbMATHGoogle Scholar
  4. Borodin OV (1995) A new proof of the 6 colour theorem. J Graph Theory 19:507–521MathSciNetCrossRefzbMATHGoogle Scholar
  5. Borodin OV (1996) Structural theorm on plane graphs with application to the entire coloring number. J Graph Theory 23:233–239MathSciNetCrossRefzbMATHGoogle Scholar
  6. Borodin OV (2013) Colorings of plane graphs: a survey. Discret Math 313:517–539MathSciNetCrossRefzbMATHGoogle Scholar
  7. Dong W (2012) A note on entire choosability of plane graphs. Discret Appl Math 160:1257–1261MathSciNetCrossRefzbMATHGoogle Scholar
  8. Hetherington TJ (2009) Entire choosablity of near-outerplane graphs. Discret Math 309:2153–2165MathSciNetCrossRefzbMATHGoogle Scholar
  9. Hu X, Wang W, Wang Y (2014) The edge-face choosability of plane graphs with maximum degree at least 9. Discret Math 327:1–8MathSciNetCrossRefzbMATHGoogle Scholar
  10. Hu X, Wang Y (2014) Plane graphs are entirely \((\Delta +5)\)-choosable. Discret Math Algorithms Appl 6(2):1450023 9 pp.MathSciNetCrossRefzbMATHGoogle Scholar
  11. Kronk HV, Mitchem J (1972) The entire chromatic number of a normal graph is at most seven. Bull Am Math Soc 78:799–800MathSciNetCrossRefzbMATHGoogle Scholar
  12. Kronk HV, Mitchem J (1973) A seven color theorem on the sphere. Discret Math 5:253–260MathSciNetCrossRefzbMATHGoogle Scholar
  13. Sanders DP, Zhao Y (2000) On the entire coloring conjecture. Can Math Bull 43:108–114MathSciNetCrossRefzbMATHGoogle Scholar
  14. Vizing VG (1964) On an estimate of the chromatic index of a p-graph. Diskret Anal 3:25–30MathSciNetGoogle Scholar
  15. Wang W (1995) On the colorings of outerplanar graphs. Discret Math 147:257–269MathSciNetCrossRefzbMATHGoogle Scholar
  16. Wang W (1999) Upper bounds of entire chromatic number of plane graphs. Eur J Comb 20:313–315MathSciNetCrossRefzbMATHGoogle Scholar
  17. Wang W, Zhu X (2011) Entire coloring of plane graphs. J Comb Theory Ser B 101:490–501CrossRefzbMATHGoogle Scholar
  18. Wang Y, Mao X, Miao Z (2013) Plane graphs with maximum degree \(\Delta \ge 8\) are entirely \((\Delta +3)\)-colorable. J Graph Theory 73:305–317MathSciNetCrossRefzbMATHGoogle Scholar
  19. Wang Y, Hu X, Wang W (2014) Plane graphs with \(\Delta \ge 9\) are entirely \((\Delta +2)\)-colorable. SIAM J Discret Math 28:1892–1905CrossRefzbMATHGoogle Scholar
  20. Wu J, Wu Y (2005) The entire coloring of series-parallel graphs. Acta Math Appl Sin Engl Ser 21:61–66MathSciNetCrossRefzbMATHGoogle Scholar
  21. Zhang Z, Wang J, Wang W, Wang L (1993) The complete chromatic number of some planar graphs. Sci China Ser A 36:1169–1177MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Weifan Wang
    • 1
    Email author
  • Tingting Wu
    • 1
  • Xiaoxue Hu
    • 1
  • Yiqiao Wang
    • 2
  1. 1.Department of MathematicsZhejiang Normal UniversityJinhuaChina
  2. 2.School of ManagementBeijing University of Chinese MedicineBeijingChina

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