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

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.