Acta Mathematica Vietnamica

, Volume 44, Issue 2, pp 493–500 | Cite as

On Chromatic Numbers of Two Extensions of Planar Graphs

  • Khosro TajbakhshEmail author


In this paper, the acyclic chromatic and the circular list chromatic numbers of a simple H-minor free graph G is considered, where H ∈{K5,K3,3}. It is proved that the acyclic chromatic number (resp. the circular list chromatic number) of a simple H-minor free graph G where H ∈{K5,K3,3} is at most 5 (resp. at most 8) and we conclude that G is star 20-colorable. These results generalize the same known results on planar graphs. Moreover, some upper bounds for the coloring numbers of H-minor free graphs for H ∈{K5,K3,3,Kr,s} and r ≤ 2 are obtained. These results generalize some known results and give some new results on group choice number, group chromatic number, and the choice number of the mentioned graphs with much shorter proofs.


Acyclic coloring Star coloring Circular choosability Coloring number Planar graphs 

Mathematics Subject Classification (2010)




The author wishes to thank Dr. G. R. Omidi for the useful comments and suggestions regarding the manuscript. I also would like to thank the anonymous referee for his/her useful comments.


  1. 1.
    Albertson, M.O., Chappell, G.G., Kierstead, H.A., Kündgen, A., Ramamurthi, R.: Coloring with no 2-colored P 4s. Electron. J. Combin. 11(1), 13 (2004)zbMATHGoogle Scholar
  2. 2.
    Borodin, O.V.: On acyclic colorings of planar graphs. Discrete Math. 25(3), 211–236 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Chuang, H., Lai, H.-J., Omidi, G.R., Zakeri, N.: On group choosability of graphs I. Ars Combin. 126, 195–209 (2016)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Chuang, H., Lai, H.-J., Omidi, G.R., Wang, K., Zakeri, N.: On group choosability of graphs II. Graphs Combin. 30(3), 549–563 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Diestel, R.: Graph Theory. 3rd edition. Springer, Berlin (2005)Google Scholar
  6. 6.
    Erdös, P., Hajnal, A.: On the chromatic number of graphs and set-systems. Acta Math. Acad. Sci. Hungar. 17(1-2), 61–99 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Fertin, G., Raspaud, A., Reed, B.: On star coloring of graphs. In: WG 2001 27th International Workshop on Graph-Theoretic Concepts in Computer Science, Springer Lecture Notes in Computer Science 2204, pp 140–153 (2001)Google Scholar
  8. 8.
    Grünbaum, B.: Acyclic colorings of planar graphs. Israel J. Math. 14, 390–408 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Havet, F., Kang, R., Müller, T., Sereni, J.-S.: Circular choosability. J. Graph Theory 61(4), 241–270 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Jaeger, F., Linial, N., Payan, C., Tarsi, M.: Group connectivity of graphs—a nonhomogeneous analogue of nowhere-zero flow properties. J. Combin. Theory Ser. B 56(2), 165–182 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Mohar, B.: Choosability for the Circular Chromatic Number. Problem of the month. (2002)
  12. 12.
    Norine, S., Wong, T.-L., Zhu, X.: Circular choosability via combinatorial Nullstellensatz. J. Graph Theory 59(3), 190–204 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Omidi, G.R.: A note on group choosability of graphs with girth at least 4. Graphs Combin. 27(2), 269–273 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Zhu, X.: Circular choosability of graphs. J. Graph Theory 48(3), 210–218 (2005)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.Department of Pure Mathematics, Faculty of Mathematical SciencesTarbiat Modares UniversityTehranIran

Personalised recommendations