A (graph) property 𝒫 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 𝒫.
If to each vertex v of a graph G a list L(v) of colors is assigned, then in an (L, 𝒫)-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 𝒫. The 𝒫-sum choice number 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, 𝒫)-coloring of G.
We state some basic results on monotonicity, give upper bounds on the 𝒫-sum choice number of arbitrary graphs for several properties, and determine the 𝒫-sum choice number of specific classes of graphs, namely, of all complete graphs, stars, paths, cycles, and all graphs of order at most 4.
SCImago Journal Rank (SJR) 2018: 0.763 Source Normalized Impact per Paper (SNIP) 2018: 0.934
Mathematical Citation Quotient (MCQ) 2017: 0.36
researchers in the fields of: colourings, partitions (general colourings), hereditary properties, independence and dominating structures (sets, paths, cycles, etc.), cycles, local properties, products of graphs