Generalized sum list colorings of graphs

A (graph) property is a class of simple finite graphs closed under isomorphisms. In this paper we consider generalizations of sum list colorings of graphs with respect to properties . If to each vertex v of a graph G a list L(v) of colors is assigned, then in an (L, )-coloring of G every vertex obtains a color from its list and the subgraphs of G induced by vertices of the same color are always in . The -sum choice number Xsc(G) $X_{sc}^\mathcal{P}(G)$ of G is the minimum of the sum of all list sizes such that, for any assignment L of lists of colors with the given sizes, there is always an (L, )-coloring of G. We state some basic results on monotonicity, give upper bounds on the -sum choice number of arbitrary graphs for several properties, and determine the -sum choice number of specific classes of graphs, namely, of all complete graphs, stars, paths, cycles, and all graphs of order at most 4.