# Generalized Fractional Total Colorings of Complete Graph

• Volume: 33, Issue: 4, page 665-676
• ISSN: 2083-5892

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## Abstract

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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 [...] 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.

## How to cite

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Gabriela Karafová. "Generalized Fractional Total Colorings of Complete Graph." Discussiones Mathematicae Graph Theory 33.4 (2013): 665-676. <http://eudml.org/doc/267837>.

@article{GabrielaKarafová2013,
abstract = {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.},
author = {Gabriela Karafová},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {fractional coloring; total coloring; complete graphs},
language = {eng},
number = {4},
pages = {665-676},
title = {Generalized Fractional Total Colorings of Complete Graph},
url = {http://eudml.org/doc/267837},
volume = {33},
year = {2013},
}

TY - JOUR
AU - Gabriela Karafová
TI - Generalized Fractional Total Colorings of Complete Graph
JO - Discussiones Mathematicae Graph Theory
PY - 2013
VL - 33
IS - 4
SP - 665
EP - 676
AB - 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.
LA - eng
KW - fractional coloring; total coloring; complete graphs
UR - http://eudml.org/doc/267837
ER -

## References

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7. [7] A. Kemnitz, M. Marangio, P. Mihók, J. Oravcová and R. Soták, Generalized fractional and circular total colorings of graphs, (2010), preprint. Zbl1317.05060
8. [8] K. Kilakos and B. Reed, Fractionally colouring total graphs, Combinatorica 13 (1993) 435-440. doi:10.1007/BF01303515[Crossref] Zbl0795.05056
9. [9] V.G. Vizing, Some unsolved problems in graph theory, Russian Math. Surveys 23 (1968) 125-141. doi:10.1070/RM1968v023n06ABEH001252 [Crossref]

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