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 [...] 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 [...] of an [...] - fractional (P,Q)-total coloring of G is called fractional (P,Q)-total chromatic number X″f,P,Q(G) = [...] 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 _X″f,P,Q(Kn) = [...] for a sufficiently large n.
@article{bwmeta1.element.doi-10_7151_dmgt_1697,
author = {Gabriela Karafov\'a},
title = {Generalized Fractional Total Colorings of Complete Graph},
journal = {Discussiones Mathematicae Graph Theory},
volume = {33},
year = {2013},
pages = {665-676},
zbl = {06323187},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_7151_dmgt_1697}
}
Gabriela Karafová. Generalized Fractional Total Colorings of Complete Graph. Discussiones Mathematicae Graph Theory, Tome 33 (2013) pp. 665-676. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_7151_dmgt_1697/
[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[Crossref] | Zbl 0902.05026
[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[Crossref] | Zbl 1234.05076
[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. | Zbl 0693.05029
[7] A. Kemnitz, M. Marangio, P. Mihók, J. Oravcová and R. Soták, Generalized fractional and circular total colorings of graphs, (2010), preprint. | Zbl 1317.05060
[8] K. Kilakos and B. Reed, Fractionally colouring total graphs, Combinatorica 13 (1993) 435-440. doi:10.1007/BF01303515[Crossref] | Zbl 0795.05056
[9] V.G. Vizing, Some unsolved problems in graph theory, Russian Math. Surveys 23 (1968) 125-141. doi:10.1070/RM1968v023n06ABEH001252 [Crossref]