Color-bounded Hypergraphs, Iii: Model Comparison

نویسندگان

  • Csilla Bujtás
  • Zsolt Tuza
چکیده

Generalizing previous models of hypergraph coloring — due to Voloshin, Drgas-Burchardt and Lazuka, and the present authors – in this paper we introduce and study the structure class that we call stably bounded hypergraphs. In this model, a hypergraph is viewed as a six-tuple H = (X, E , s, t,a, b), where s, t,a, b : E → N are given integer-valued functions on the edge set. A mapping φ : X → N is a proper vertex coloring if it satisfies the following conditions for each edge E ∈ E : the number of colors in E is at least s(E) and at most t(E), while the largest number of vertices having the same color inside E is at least a(E) and at most b(E). Taking different subsets of {s, t,a, b} (as combinations of nontrivial conditions on colorability) result in a hierarchy of structure classes with respect to vertex coloring. The main issue of this paper is to carry out a detailed analysis of how those classes are related. This includes the study of possible chromatic polynomials and ‘feasible sets’ — that is, the set Φ(H) of integers k such that H has a proper vertex coloring with exactly k colors — with or without assuming that the number of vertices is the same under the different combinations of color-bound conditions, or restricting the edge sizes. Furthermore, substantial change is observed concerning the algorithmic complexity of recognizing hypergraphs that are uniquely colorable and Φ(H) = { |X| − 1}.

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تاریخ انتشار 2007