(P, Q)-Total (r, s)-colorings of graphs

Let r, s is an element of 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 ss-element subsets of Z(r) such that for each color i, 0 <= i <= r-1, the vertices colored by subsets containing ii induce a subgraph of G with property P, the edges colored by subsets containing ii 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 chi ''(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. In this paper we present general lower and upper bounds for chi ''(f,P,Q)(G) and also give some exact values for specific properties and specific classes of graphs. (C) 2014 Elsevier B.V. All rights reserved.