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 
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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[1] M. Behzad, Graphs and their chromatic numbers, Doctoral Thesis (Michigan state University, 1965).

[2] M. Behzad, The total chromatic number of a graph, in: Combinatorial Mathematics and its Applications, D.J.A.Welsh, Ed., (Academic Press, London, 1971) 1-10.

[3] M. Borowiecki, I. Broere, M. Frick, P. Mihók and G. Semanišin, A survey of hereditary properties of graphs, Discuss. Math. Graph Theory 17 (1997) 5-50. doi:10.7151/dmgt.1037

[4] M. Borowiecki, A. Kemnitz, M. Marangio and P. Mihók, Generalized total colorings of graphs, Discuss. Math. Graph Theory 31 (2011) 209-222. doi:10.7151/dmgt.1540

[5] M. Borowiecki and P. Mihók, Hereditary properties of graphs, in: Advances in Graph Theory, V.R. Kulli, Ed., (Vishwa International Publication, Gulbarga, 1991) 41-68.

[6] A. Chetwynd, Total colourings, in: Graphs Colourings, Pitman Research Notes in Mathematics No.218, R. Nelson and R.J. Wilson Eds., (London, 1990) 65-77.

[7] A. Kemnitz, M. Marangio, P. Mihók, J. Oravcová and R. Soták, Generalized fractional and circular total colorings of graphs, (2010), preprint.

[8] K. Kilakos and B. Reed, Fractionally colouring total graphs, Combinatorica 13 (1993) 435-440. doi:10.1007/BF01303515

[9] V.G. Vizing, Some unsolved problems in graph theory, Russian Math. Surveys 23 (1968) 125-141. doi:10.1070/RM1968v023n06ABEH001252