Generalized total colorings of graphs

An additive hereditary property of graphs is a class of simple graphs which is closed under unions, subgraphs and isomorphism. Let P and Q be additive hereditary properties of graphs. A (P ,Q)-total coloring ∗Research supported in part by Slovak VEGA Grant 2/0194/10. 210 M. Borowiecki, A. Kemnitz, M. Marangio and P. Mihók of a simple graphG is a coloring of the vertices V (G) and edgesE(G) of G such that for each color i the vertices colored by i induce a subgraph of property P , the edges colored by i induce a subgraph of property Q and incident vertices and edges obtain different colors. In this paper we present some general basic results on (P ,Q)-total colorings. We determine the (P ,Q)-total chromatic number of paths and cycles and, for specific properties, of complete graphs. Moreover, we prove a compactness theorem for (P ,Q)-total colorings.