## Generalized Fractional Total Colorings of Complete Graphs

 Gabriela Karafová Institute of Mathematics, P.J. Šafárik University, Jesenná 5, 040 01 Košice, Slovakia

## Abstract

An additive and hereditary property of graphs is a class of simple graphs which is closed under unions, subgraphs and isomorphism. Let P and Q be two additive and hereditary graph properties and let r,s be integers such that r ≥ s. Then an (r/s)-fractional (P,Q)-total coloring of a finite graph G = (V,E) is a mapping f, which assigns an s-element subset of the set {1,2,...,r} to each vertex and each edge, moreover, for any color i all vertices of color i induce a subgraph of property P, all edges of color i induce a subgraph of property Q and vertices and incident edges have assigned disjoint sets of colors. The minimum ratio r/s of an (r/s)-fractional (P,Q)-total coloring of G is called fractional (P,Q)-total chromatic number χ ′ ′f,P,Q(G) = r/s. Let k = sup{i:Ki+1P} and l = sup{i:Ki+1Q}. We show for a complete graph Kn that if l ≥ k+2 then χ ′ ′f,P,Q(Kn) = n/(k+1) for a sufficiently large n.

Keywords: fractional coloring, total coloring, complete graphs

2010 Mathematics Subject Classification: 05C15.

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