GENERALIZED FRACTIONAL TOTAL COLORINGS OF COMPLETE GRAPHS

An additive and hereditary property of graphs is a class of simple graphs which is closed under unions, subgraphs and isomorphism. Let P and 2 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 L's of an,"ii-fractional (P, Q)-total coloring of G is called fractional (P, Q)-total chromatic number x ''(f.P,Q)(G) = r/s. Let k = sup{i : Ki+1 is an element of P} and l = sup{i : Ki+1 is an element of Q}. We show for a complete graph Kn, that if l >= k + 2 then x ''(f.P,Q)(K-n) -n/k+1 for a sufficiently large n.