Generalized fractional and circular total colorings of graphs

Let P and Q be additive and hereditary graph properties, r, s ∈ N, r ≥ s, and [Zr] be the set of all s-element subsets of Zr. An (r, s)-fractional This work was supported by the Slovak Science and Technology Assistance Agency under the contract No. APVV-0023-10 and by the Slovak VEGA Grant 1/0652/12. 2 A. Kemnitz, M. Marangio, P. Mihók, J. Oravcová and R. Soták (P,Q)-total coloring of G is an assignment h : V (G) ∪ E(G) → [Zr] such that for each i ∈ Zr the following holds: the vertices of G whose color sets contain color i induce a subgraph of G with property P, edges with color sets containing color i induce a subgraph of G with property Q, and the color sets of incident vertices and edges are disjoint. If each vertex and edge of G is colored with a set of s consecutive elements of Zr we obtain an (r, s)-circular (P,Q)-total coloring of G. In this paper we present basic results on (r, s)-fractional/circular (P,Q)-total colorings. We introduce the fractional and circular (P,Q)-total chromatic number of a graph and we determine this number for complete graphs and some classes of additive and hereditary properties.