(P, Q)-Total (r, s)-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 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 ? 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 ? f , P , Q ? ( G ) and also give some exact values for specific properties and specific classes of graphs.