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

Mojtaba Abdolmaleki, Joan P. Hutchinson, S. Gh. Ilchi, Ebadollah S. Mahmoodian, Naoki Matsumoto, Mohammad Amin Shabani

Abstract: A graph G is called uniquely k -list colorable (UkLC) if there exists a list of colors on its vertices, say $$L=\lbrace S_v \mid v \in V(G) \rbrace $$ L = { S v ∣ v ∈ V ( G ) } , 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 $$k\ge 3$$ k ≥ 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.