Fractional ({ P},{ Q})-total list colorings of graphs

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 Zr 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 (P,Q)-total independent set T = VT ∪ET ⊆ V ∪E is the union of a set VT of vertices and a set ET of edges of G such that for the graphs induced by the sets VT and ET it holds that G[VT ] ∈ P, G[ET ] ∈ Q, and G[VT ] and G[ET ] are disjoint. Let TP,Q be the set of all (P,Q)-total independent sets of G. Let L(x) be a set of admissible colors for every element x ∈ V ∪ E. The graph G is called (P,Q)-total (a, b)-list colorable if for each list assignment L with |L(x)| = a for all x ∈ V ∪E it is possible to choose a subset C(x) ⊆ L(x) with |C(x)| = b for all x ∈ V ∪ E such that the set Ti which is defined by Ti = {x ∈ V ∪ E : i ∈ C(x)} belongs to TP,Q for every color i. The (P,Q)- choice ratio chrP,Q(G) of G is defined as the infimum of all ratios a/b such that G is (P,Q)-total (a, b)-list colorable. We give a direct proof of χ′′ f,P,Q(G) = chrP,Q(G) for all simple graphs G and we present for some properties P and Q new bounds for the (P,Q)-total chromatic number and for the (P,Q)-choice ratio of a graph G.