Generalized total colorings of graphs

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 * Research supported in part by Slovak VEGA Grant 2/0194/10. 210M. Borowiecki, A. Kemnitz, M. Marangio and P. Mihók 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.

Some bounds on the generalised total chromatic number of degenerate graphs

“…The minimum number of colours needed in a total (P, Q)-colouring of G is called the total (P, Q)-chromatic number and is denoted by χ P,Q (G) (see [2]). Clearly, when P = O and Q = O 1 , a total (P, Q)-colouring of a graph G is nothing but a total colouring of G so that χ O,O 1 (G) = χ (G).…”

Fractional ({ P},{ Q})-total list colorings of graphs

“…The fractional (P, Q)-total chromatic number χ ′′ f,P,Q (G) of G is defined by (P, Q)-total colorings were introduced in [7] and (P, Q)-total (r, s)-colorings in [11] where first results can be found. For example, it was shown in [11] that the following definition of the fractional (P, Q)-total chromatic number is equivalent to the one given above.…”

Fractional Q-coloring of graphs

“…We list several well-known additive hereditary properties Generalized colorings of edges or/and vertices of graphs under restrictions given by graph properties have recently attracted much attention, see e.g. [2,3,4,6,7,8,10] and references therein.…”