Choosability with Separation of Cycles and Outerplanar Graphs

We consider the following list coloring with separation problem of graphs. Given a graph G and integers a, b, find the largest integer c such that for any list assignment L of G with |L(v)| <= a for any vertex v and |L(u) boolean AND L(v)| <= c for any edge uv of G, there exists an assignment phi of sets of integers to the vertices of G such that phi(u) subset of L(u) and |phi(v)| = b for any vertex v and phi(u) boolean AND phi(v) = null for any edge uv. Such a value of c is called the separation number of (G, a, b). We also study the variant called the free-separation number which is defined analogously but assuming that one arbitrary vertex is precolored. We determine the separation number and free-separation number of the cycle and derive from them the free-separation number of a cactus. We also present a lower bound for the separation and free-separation numbers of outerplanar graphs of girth g >= 5.