Let r, s ∈ N, r ≥ s, and P and Q be two additive and hereditary graph properties. A (P, Q)-total (r, s)-coloring of a graph G = (V, E) is a coloring of the vertices and edges of G by s-element subsets of Z r such that for each color i, 0 ≤ i ≤ r − 1, the vertices colored by subsets containing i induce a subgraph of G with property P, the edges colored by subsets containing i induce a subgraph of G with property Q, and color sets of incident vertices and edges are disjoint. The fractional (P, Q)-total chromatic number χ ′′ f,P,Q (G) of G is defined as the infimum of all ratios r/s such that G has a (P, Q)-total (r, s)-coloring.
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 χ P sc (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, P)-coloring of G. We state some basic results on monotonicity, give upper bounds on the Psum 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.
We consider the problem of the existence of uniquely partitionable planar graphs. We survey some recent results and we prove the nonexistence of uniquely (D 1 ,D 1 )-partitionable planar graphs with respect to the property D 1 "to be a forest".
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