Graphs and Combinatorics

, Volume 34, Issue 3, pp 383–394 | Cite as

On Uniquely k-List Colorable Planar Graphs, Graphs on Surfaces, and Regular Graphs

  • M. Abdolmaleki
  • J. P. Hutchinson
  • S. Gh. Ilchi
  • E. S. Mahmoodian
  • N. Matsumoto
  • M. A. Shabani
Original Paper
  • 52 Downloads

Abstract

A graph G is called uniquely k -list colorable (UkLC) if there exists a list of colors on its vertices, say \(L=\lbrace S_v \mid v \in V(G) \rbrace \), each of size k, such that there is a unique proper list coloring of G from this list of colors. A graph G is said to have property M(k) if it is not uniquely k-list colorable. Mahmoodian and Mahdian (Ars Comb 51:295–305, 1999) characterized all graphs with property M(2). For \(k\ge 3\) property M(k) has been studied only for multipartite graphs. Here we find bounds on M(k) for graphs embedded on surfaces, and obtain new results on planar graphs. We begin a general study of bounds on M(k) for regular graphs, as well as for graphs with varying list sizes.

Keywords

Uniquely list colorable graphs Planar graphs Regular graphs Graphs on surfaces 

Notes

Acknowledgements

We thank the anonymous referee for her/his careful reading of our manuscript and her/his many insightful comments and suggestions. Part of the research of E. S. M. was supported by INSF and the Research Office of the Sharif University of Technology.

References

  1. 1.
    Bondy, J.A., Murty, U.S.R.: Graph theory, Graduate Texts in Mathematics, vol. 244. Springer, New York (2008).  https://doi.org/10.1007/978-1-84628-970-5
  2. 2.
    Dinitz, J.H., Martin, W.J.: The stipulation polynomial of a uniquely list-colorable graph. Australas. J. Comb. 11, 105–115 (1995)MathSciNetMATHGoogle Scholar
  3. 3.
    Eslahchi, C., Ghebleh, M., Hajiabolhassan, H.: Some concepts in list coloring. J. Comb. Math. Comb. Comput. 41, 151–160 (2002)MathSciNetMATHGoogle Scholar
  4. 4.
    Ganjali, Y.G., Ghebleh, M., Hajiabolhassan, H., Mirzazadeh, M., Sadjad, B.S.: Uniquely 2-list colorable graphs. Discrete Appl. Math. 119(3), 217–225 (2002).  https://doi.org/10.1016/S0166-218X(00)00335-8
  5. 5.
    Ghebleh, M., Mahmoodian, E.S.: On uniquely list colorable graphs. Ars Comb. 59, 307–318 (2001)MathSciNetMATHGoogle Scholar
  6. 6.
    He, W., Shen, Y., Zhao, Y., Wang, Y., Ma, X.: On property \(M(3)\) of some complete multipartite graphs. Australas. J. Comb. 35, 211–220 (2006)MathSciNetMATHGoogle Scholar
  7. 7.
    Jensen, T.R., Toft, B.: Graph coloring problems. Wiley-Interscience Series in Discrete Mathematics and Optimization. A Wiley-Interscience Publication. Wiley, New York (1995)Google Scholar
  8. 8.
    Mahdian, M., Mahmoodian, E.S.: A characterization of uniquely \(2\)-list colorable graphs. Ars Comb. 51, 295–305 (1999)MathSciNetMATHGoogle Scholar
  9. 9.
    Mahmoodian, E.S., Mahdian, M.: On the uniquely list colorable graphs. In: Proceedings of the 28th Annual Iranian Mathematics Conference, Part 1 (Tabriz, 1997), Tabriz Univ. Ser., vol. 377, pp. 319–326. Tabriz Univ., Tabriz (1997)Google Scholar
  10. 10.
    Marx, D.: Complexity of unique list colorability. Theor. Comput. Sci. 401(1-3), 62–76 (2008).  https://doi.org/10.1016/j.tcs.2008.03.018
  11. 11.
    Shen, Y., Wang, Y., He, W., Zhao, Y.: On uniquely list colorable complete multipartite graphs. Ars Comb. 88, 367–377 (2008)MathSciNetMATHGoogle Scholar
  12. 12.
    Wang, Y., Shen, Y., Zheng, G., He, W.: On uniquely 4-list colorable complete multipartite graphs. Ars Comb. 93, 203–214 (2009)MathSciNetMATHGoogle Scholar
  13. 13.
    Wang, Y., Wang, Y., Zhang, X.: Some conclusion on unique \(k\)-list colorable complete multipartite graphs. J. Appl. Math., Art. ID 380,861, p. 5 (2013).  https://doi.org/10.1155/2013/380861
  14. 14.
    Zhao, Y., He, W., Shen, Y., Wang, Y.: Note on characterization of uniquely 3-list colorable complete multipartite graphs. In: Discrete geometry, combinatorics and graph theory, Lecture Notes in Comput. Sci., vol. 4381, pp. 278–287. Springer, Berlin (2007).  https://doi.org/10.1007/978-3-540-70666-3_30

Copyright information

© Springer Japan KK, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematical SciencesSharif University of TechnologyTehranIslamic Republic of Iran
  2. 2.Department of Mathematics, Statistics, and Computer ScienceMacalester CollegeSaint PaulUSA
  3. 3.Department of Computer EngineeringSharif University of TechnologyTehranIslamic Republic of Iran
  4. 4.Department of Environment and Information SciencesYokohama National UniversityYokohamaJapan

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