(P,Q)-Total (r,s)

Abstract

a b s t r a c tLet 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.In this paper we present general lower and upper bounds for χ ′′ f ,P ,Q (G) and also give some exact values for specific properties and specific classes of graphs.