## GENERALIZED TOTAL COLORINGS OF GRAPHS

Mieczysław Borowiecki

Faculty of Mathematics, Computer Science and Econometrics
University of Zielona Góra
prof. Z. Szafrana 4a, 65-516 Zielona Góra, Poland
e-mail: m.borowiecki@wmie.uz.zgora.pl

Arnfried Kemnitz, Massimiliano Marangio

Computational Mathematics
Technische Universität Braunschweig
Pockelsstr. 14, 38106 Braunschweig, Germany
 e-mail: m.marangio@tu-bs.de e-mail: a.kemnitz@tu-bs.de

Peter Mihók

Department of Applied Mathematics and Informatics
Faculty of Economics, Technical University of Košice
B. Nemcovej 32, 04001 Košice, and
Mathematical Institute of Slovak Academy of Sciences
Gresákova 6, 04001 Košice, Slovakia
e-mail: peter.mihok@tuke.sk

## Abstract

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 of a simple graph G is a coloring of the vertices V(G) and edges E(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.

Keywords: hereditary properties, generalized total colorings, paths, cycles, complete graphs.

2010 Mathematics Subject Classification: 05C15.

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