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

graph G is called uniquely k -list colorable (U k LC) if there exists a list of colors on its vertices, say 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≥ 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.