On Uniquely k-List Colorable Planar Graphs, Graphs on Surfaces, and Regular Graphs
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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.
KeywordsUniquely list colorable graphs Planar graphs Regular graphs Graphs on surfaces
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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