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Arnfried Kemnitz, Peter Mihók and Margit Voigt

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.

Open access

Arnfried Kemnitz, Massimiliano Marangio and Margit Voigt

Abstract

Let G = (V,E) be a simple graph and for every edge e ∈ E let L(e) be a set (list) of available colors. The graph G is called L-edge colorable if there is a proper edge coloring c of G with c(e) ∈ L(e) for all e ∈ E. A function f : E → ℕ is called an edge choice function of G and G is said to be f-edge choosable if G is L-edge colorable for every list assignment L with |L(e)| = f(e) for all e ∈ E. Set size(f) = ∑e∈E f(e) and define the sum choice index χ′sc(G) as the minimum of size(f) over all edge choice functions f of G.

There exists a greedy coloring of the edges of G which leads to the upper bound χ′sc(G) ≤ 1/2 ∑v∈V d(v)2. A graph is called sec-greedy if its sum choice index equals this upper bound.

We present some general results on the sum choice index of graphs including a lower bound and we determine this index for several classes of graphs. Moreover, we present classes of sec-greedy graphs as well as all such graphs of order at most 5.

Open access

Arnfried Kemnitz, Massimiliano Marangio and Margit Voigt

Abstract

A (graph) property 𝒫 is a class of simple finite graphs closed under isomorphisms. In this paper we consider generalizations of sum list colorings of graphs with respect to properties 𝒫.

If to each vertex v of a graph G a list L(v) of colors is assigned, then in an (L, 𝒫)-coloring of G every vertex obtains a color from its list and the subgraphs of G induced by vertices of the same color are always in 𝒫. The 𝒫-sum choice number Xsc𝒫(G) of G is the minimum of the sum of all list sizes such that, for any assignment L of lists of colors with the given sizes, there is always an (L, 𝒫)-coloring of G.

We state some basic results on monotonicity, give upper bounds on the 𝒫-sum choice number of arbitrary graphs for several properties, and determine the 𝒫-sum choice number of specific classes of graphs, namely, of all complete graphs, stars, paths, cycles, and all graphs of order at most 4.