Fractional (P,Q)-Total List Colorings of Graphs
Arnfried Kemnitz ; Peter Mihók ; Margit Voigt
Discussiones Mathematicae Graph Theory, Tome 33 (2013), p. 167-179 / Harvested from The Polish Digital Mathematics Library

Let r, s ∈ N, r ≥ s, and P and Q be two additive and hereditary graph properties. A (P,Q)-total (r, s)-coloring of a graph G = (V,E) is a coloring of the vertices and edges of G by s-element subsets of Zr such that for each color i, 0 ≤ i ≤ r − 1, the vertices colored by subsets containing i induce a subgraph of G with property P, the edges colored by subsets containing i induce a subgraph of G with property Q, and color sets of incident vertices and edges are disjoint. The fractional (P,Q)-total chromatic number χ′′ f,P,Q(G) of G is defined as the infimum of all ratios r/s such that G has a (P,Q)-total (r, s)-coloring. A (P,Q)-total independent set T = VT ∪ET ⊆ V ∪E is the union of a set VT of vertices and a set ET of edges of G such that for the graphs induced by the sets VT and ET it holds that G[VT ] ∈ P, G[ET ] ∈ Q, and G[VT ] and G[ET ] are disjoint. Let TP,Q be the set of all (P,Q)-total independent sets of G. Let L(x) be a set of admissible colors for every element x ∈ V ∪ E. The graph G is called (P,Q)-total (a, b)-list colorable if for each list assignment L with |L(x)| = a for all x ∈ V ∪E it is possible to choose a subset C(x) ⊆ L(x) with |C(x)| = b for all x ∈ V ∪ E such that the set Ti which is defined by Ti = {x ∈ V ∪ E : i ∈ C(x)} belongs to TP,Q for every color i. The (P,Q)- choice ratio chrP,Q(G) of G is defined as the infimum of all ratios a/b such that G is (P,Q)-total (a, b)-list colorable. We give a direct proof of χ′′ f,P,Q(G) = chrP,Q(G) for all simple graphs G and we present for some properties P and Q new bounds for the (P,Q)-total chromatic number and for the (P,Q)-choice ratio of a graph G.

Publié le : 2013-01-01
EUDML-ID : urn:eudml:doc:267990
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     title = {Fractional (P,Q)-Total List Colorings of Graphs},
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     volume = {33},
     year = {2013},
     pages = {167-179},
     zbl = {1291.05068},
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Arnfried Kemnitz; Peter Mihók; Margit Voigt. Fractional (P,Q)-Total List Colorings of Graphs. Discussiones Mathematicae Graph Theory, Tome 33 (2013) pp. 167-179. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_7151_dmgt_1649/

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