Discussiones Mathematicae Graph Theory 33(1) (2013)
167-179
DOI: https://doi.org/10.7151/dmgt.1649
Dedicated to Mieczysław Borowiecki on the occasion of his 70th birthday
Fractional (P,Q)-total List Colorings of Graphs
Arnfried Kemnitz
Computational Mathematics, Technical University Braunschweig | Peter Mihók Mathematical Institute | Margit Voigt
Faculty of Information Technology and Mathematics, University of Applied Sciences |
Abstract
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.
Keywords: graph property, total coloring, (P,Q)-total coloring, fractional coloring, fractional (P,Q)-total chromatic number, circular coloring, circular (P,Q)-total chromatic number, list coloring, (P,Q)-total (a,b)-list colorings
2010 Mathematics Subject Classification: 05C15, 05C75.
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Received 19 March 2012
Revised 23 August 2012
Accepted 23 August 2012
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