Skip to main content
SpringerLink
Log in

Equitable List-Coloring for C 5-Free Plane Graphs Without Adjacent Triangles

  • Original Paper
  • Published:
Graphs and Combinatorics Aims and scope Submit manuscript

Abstract

A graph G is equitably k-choosable if, for any k-uniform list assignment L, G is L-colorable and each color appears on at most \({\lceil\frac{|V(G)|}{k}\rceil}\) vertices. Kostochka, Pelsmajer and West introduced this notion and conjectured that G is equitably k-choosable for \({k > \Delta(G)}\). In this paper, we prove that every C 5-free plane graph G without adjacent triangles is equitably k-choosable whenever \({k \geq \max\{\Delta(G), 8\}}\).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Chen B.L., Lih K.W., Wu P.L.: Equitable coloring and the maximum degree. Eur. J. Combin. 15, 443–447 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  2. Chen B.L., Lih K.W.: Equitable coloring of trees. J. Combin. Theory Ser. B 61, 83–87 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  3. Dong A., Zhang X., Li G.J.: Equitable coloring and equitable choosability of planar graphs without 5- and 7-cycles. Bull. Malays. Math. Sci. Soc. 35, 897C910 (2012)

    MathSciNet  Google Scholar 

  4. Hajnal, A., Szemerédi, E.: Proof of a conjecture of Erdös. In: Rényi, A., Sós, V.T. (eds.) Combin Theory and its applications,Vol II, pp. 601–623. North-Holland,Amsterdam (1970)

  5. Kierstead H.A., Kostochka A.V.: A short proof of the Hajnal-Szemerdi Theorem on equitable coloring, Combinatorics. Probab. Comput. 17, 265C270 (2008)

    MathSciNet  Google Scholar 

  6. Kierstead H.A., Kostochka A.V.: Every 4-colorable graph with maximum degree 4 has an equitable 4-coloring. J. Graph Theory 71, 31C48 (2012)

    Article  MathSciNet  Google Scholar 

  7. Kierstead, H.A.; Kostochka, A.V.: Equitable list coloring of graphs with bounded degree. J. Graph Theory. doi:10.1002/jgt.21710

  8. Kierstead, H.A., Kostochka, A.V.: A refinement of a result of Corrádi and Hajnal, accepted

  9. Kostochka A.V., Nakprasit K.: On equitable \({\Delta}\) -coloring of graphs with low average degree. Theor. Comp. Sci. 349, 82C91 (2005)

    Article  MathSciNet  Google Scholar 

  10. Kostochka A.V., Pelsmajer M.J., West D.B.: A list analogue of equitable coloring. J. Graph Theory 44, 166–177 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  11. Li Q., Bu Y.H.: Equitable list coloring of planar graphs without 4- and 6-cycles. Discrete Math. 309, 280C287 (2009)

    MathSciNet  Google Scholar 

  12. Lih K.W., Wu P.L.: On equitable coloring of bipartite graphs. Discrete Math. 151, 155–160 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  13. Meyer W.: Equitable coloring. Amer. Math. Monthly 80, 920–922 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  14. Nakprasit K.: Equitable colorings of planar graphs with maximum degree at least nine. Discrete Math. 312, 1019–1024 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  15. Nakprasit K., Nakprasit K.: Equitable colorings of planar graphs without short cycles. Theoret. Comput. Sci. 465, 21–27 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  16. Pelsmajer M.J.: Equitable list-coloring for graphs of maximum degree 3. J. Graph Theroy 47, 1–8 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  17. Wang W.F., Lih K.W.: Choosability and edge choosability of planar graphs without 5-cycles. Appl. Math. Lett. 15, 561–565 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  18. Wang W.F., Lih K.W.: Equitable list coloring for graphs. Taiwanese J. Math. 8, 747–759 (2004)

    MATH  MathSciNet  Google Scholar 

  19. Wang W.F., Zhang K.M.: Equitable colorings of line graphs and complete r-partite graphs. System Sci. Math. Sci. 13, 190–194 (2000)

    MATH  Google Scholar 

  20. Yap H.P., Zhang Y.: The equitable \({\Delta}\) -coloring conjecture holds for outerplanar graphs, Bull. Inst. Math. Acad. Sinica 25, 143–149 (1997)

    MATH  MathSciNet  Google Scholar 

  21. Yap H.P., Zhang Y.: Equitable colorings of planar graphs. J. Combin. Math. Combin. Comput. 27, 97–105 (1998)

    MATH  MathSciNet  Google Scholar 

  22. Zhu J.L., Bu Y.H.: Equitable list colorings of planar graphs without short cycles. Theoret. Comput. Sci. 407, 21–28 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  23. Zhu J.L., Bu Y.H.: Equitable and equitable list colorings of graphs. Theoret. Comput. Sci. 411, 3873–3876 (2010)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Jiaxing University, Jiaxing, 314001, People’s Rupublic of China

    Junlei Zhu & Xiao Min

  2. Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, People’s Republic of China

    Yuehua Bu

Authors
  1. Junlei Zhu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yuehua Bu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xiao Min
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Junlei Zhu.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zhu, J., Bu, Y. & Min, X. Equitable List-Coloring for C 5-Free Plane Graphs Without Adjacent Triangles. Graphs and Combinatorics 31, 795–804 (2015). https://doi.org/10.1007/s00373-013-1396-7

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00373-013-1396-7

Keywords

Mathematics Subject Classification (2000)

Search

Navigation