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arXiv · 1705.07434

On uniquely k-list colorable planar graphs, graphs on surfaces, and regular graphs

M. Abdolmaleki·J. P. Hutchinson·S. Gh. Ilchi·E. S. Mahmoodian·M. A. Shabani
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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 characterized all graphs with property $M(2)$. For $k\geq 3$ 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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M. Abdolmaleki, J. P. Hutchinson, S. Gh. Ilchi, E. S. Mahmoodian, M. A. Shabani. 2017-05-21. On uniquely k-list colorable planar graphs, graphs on surfaces, and regular graphs. https://arxiv.org/abs/1705.07434

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