Color-bounded hypergraphs, I: General results

Csilla Bujtás, Zsolt Tuza

Research output: Contribution to journalArticle

17 Citations (Scopus)

Abstract

The concept of color-bounded hypergraph is introduced here. It is a hypergraph (set system) with vertex set X and edge set E = {E1, ..., Em}, where each edge Ei is associated with two integers si and ti such that 1 ≤ si ≤ ti ≤ | Ei |. A vertex coloring φ : X → N is considered to be feasible if the number of colors occurring in Ei satisfies si ≤ | φ (Ei) | ≤ ti, for all i ≤ m. Color-bounded hypergraphs generalize the concept of 'mixed hypergraphs' introduced by Voloshin [V. Voloshin, The mixed hypergraphs, Computer Science Journal of Moldova 1 (1993) 45-52], and a recent model studied by Drgas-Burchardt and Łazuka [E. Drgas-Burchardt, E. Łazuka, On chromatic polynomials of hypergraphs, Applied Mathematics Letters 20 (12) (2007) 1250-1254] where only lower bounds si were considered. We discuss the similarities and differences between our general model and the more particular earlier ones. An important issue is the chromatic spectrum-strongly related to the chromatic polynomial-which is the sequence whose kth element is the number of allowed colorings with precisely k colors (disregarding color permutations). Problems concerning algorithmic complexity are also considered.

Original languageEnglish
Pages (from-to)4890-4902
Number of pages13
JournalDiscrete Mathematics
Volume309
Issue number15
DOIs
Publication statusPublished - Aug 6 2009

Keywords

  • Chromatic polynomial
  • Feasible set
  • Hypergraph
  • Mixed hypergraph
  • Uniquely colorable hypergraph
  • Vertex coloring

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics

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