On Uniquely k-List Colorable Planar Graphs, Graphs on Surfaces, and Regular Graphs (original) (raw)

Abstract

A graph G is called uniquely k -list colorable (U_k_LC) 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](/article/10.1007/s00373-018-1879-7#ref-CR8 "Mahdian, M., Mahmoodian, E.S.: A characterization of uniquely 

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            -list colorable graphs. 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.

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Notes

  1. The complexity class \(\sum _{2}^{p} = NP^{NP}\) contains those problems that can be solved by a polynomial-time nondeterministic Turing machine equipped with an NP-oracle [10].

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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.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Sharif University of Technology, P.O. Box 11155-9415, Tehran, Islamic Republic of Iran
    M. Abdolmaleki, E. S. Mahmoodian & M. A. Shabani
  2. Department of Mathematics, Statistics, and Computer Science, Macalester College, Saint Paul, MN, USA
    J. P. Hutchinson
  3. Department of Computer Engineering, Sharif University of Technology, P.O. Box 11155-9415, Tehran, Islamic Republic of Iran
    S. Gh. Ilchi
  4. Department of Environment and Information Sciences, Yokohama National University, 79-7 Tokiwadai, Hodogaya-ku, Yokohama, 240-8501, Japan
    N. Matsumoto

Authors

  1. M. Abdolmaleki
  2. J. P. Hutchinson
  3. S. Gh. Ilchi
  4. E. S. Mahmoodian
  5. N. Matsumoto
  6. M. A. Shabani

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Correspondence toE. S. Mahmoodian.

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Abdolmaleki, M., Hutchinson, J.P., Ilchi, S.G. et al. On Uniquely _k_-List Colorable Planar Graphs, Graphs on Surfaces, and Regular Graphs.Graphs and Combinatorics 34, 383–394 (2018). https://doi.org/10.1007/s00373-018-1879-7

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