Geodesic
Generalized Fractional and Circular Total Colorings of Graphs
Discussiones Mathematicae. Graph Theory, Tome 35 (2015) no. 3, pp. 517-532

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Let 𝒫 and 𝒬 be additive and hereditary graph properties, r, s ∈ℕ, r ≥ s, and [ℤ_r]^s be the set of all s-element subsets of ℤ_r. An (r, s)-fractional (𝒫,𝒬)-total coloring of G is an assignment h : V (G) ∪ E(G) → [ℤ_r]^s such that for each i ∈ℤ_r the following holds: the vertices of G whose color sets contain color i induce a subgraph of G with property 𝒫, edges with color sets containing color i induce a subgraph of G with property 𝒬, and the color sets of incident vertices and edges are disjoint. If each vertex and edge of G is colored with a set of s consecutive elements of ℤ_r we obtain an (r, s)-circular (𝒫,𝒬)-total coloring of G. In this paper we present basic results on (r, s)-fractional/circular (𝒫,𝒬)-total colorings. We introduce the fractional and circular (𝒫,𝒬)-total chromatic number of a graph and we determine this number for complete graphs and some classes of additive and hereditary properties.
Keywords: graph property, (P,Q)-total coloring, circular coloring, fractional coloring, fractional (P,Q)-total chromatic number, circular (P,Q)- total chromatic number 
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     author = {Kemnitz, Arnfried and Marangio, Massimiliano and Mih\'ok, Peter and Oravcov\'a, Janka and Sot\'ak, Roman},
     title = {Generalized {Fractional} and {Circular} {Total} {Colorings} of {Graphs}},
     journal = {Discussiones Mathematicae. Graph Theory},
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     year = {2015},
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Kemnitz, Arnfried; Marangio, Massimiliano; Mihók, Peter; Oravcová, Janka; Soták, Roman. Generalized Fractional and Circular Total Colorings of Graphs. Discussiones Mathematicae. Graph Theory, Tome 35 (2015) no. 3, pp. 517-532. http://geodesic.mathdoc.fr/item/DMGT_2015_35_3_a9/