Generalized fractional total coloring of complete graphs
Gabriela Karafová · Discussiones Mathematicae Graph Theory · 2013
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 : K i+1 ∈ P} and l = sup{i : K i+1 ∈ Q}.We show for a complete graph K n that if l ≥ k + 2 then χ ′′ f,P,Q (K n ) = n k+1 for a sufficiently large n.