Geodesic
Generalized Fractional Total Colorings of Complete Graph
Discussiones Mathematicae. Graph Theory, Tome 33 (2013) no. 4, pp. 665-676.

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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.
Keywords: fractional coloring, total coloring, complete graphs 
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Karafová, Gabriela. Generalized Fractional Total Colorings of Complete Graph. Discussiones Mathematicae. Graph Theory, Tome 33 (2013) no. 4, pp. 665-676. http://geodesic.mathdoc.fr/item/DMGT_2013_33_4_a3/

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