Generalized fractional total coloring 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 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 graphG = (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+1 ∈ P} and l = sup{i : Ki+1 ∈ Q}. 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.