C-perfect hypergraphs

Csilla Bujtás, Zsolt Tuza

Research output: Contribution to journalArticle

5 Citations (Scopus)

Abstract

Let ℋ = (X, ℰ) be a hypergraph with vertex set X and edge set ℰ. A C-coloring of ℋ is a mapping φ:X→ ℕ such that |φ(E)|<|E| holds for all edges E∈Escr; (i.e. no edge is multicolored). We denote by x(ℋ)the maximum number |φ(X)| of colors in a C-coloring. Let further α(ℋ) denote the largest cardinality of a vertex set S⊆ Xthat contains no E∈ ℰ,and τ(ℋ) = |X|-α(ℋ) the minimum cardinality of a vertex set meeting all E∈ℰ. The hypergraph ℋ is called C-perfect if x(ℋ') = α(ℋ') holds for every induced subhypergraph ℋ ⊆ℋ.lf ℋ is not C-perfect but all of its proper induced subhypergraphs are, then we say that it is minimally C-imperfect. We prove that for all r, k∈ ℕ there exists a finite upper bound h(r, k)onthe number of minimally C-imperfect hypergraphs ℋ with τ(ℋ) ≤ k and without edges of more than r vertices. We give a characterization of minimally C-imperfect hypergraphs that have τ = 2, which also characterizes implicitly the C-perfect ones with τ = 2. From this result we derive an infinite family of new constructions that are minimally C-imperfect. A characterization of minimally C-imperfect circular hypergraphs is presented, too.

Original languageEnglish
Pages (from-to)132-149
Number of pages18
JournalJournal of Graph Theory
Volume64
Issue number2
DOIs
Publication statusPublished - Jun 1 2010

    Fingerprint

Keywords

  • C-perfect hypergraph
  • Circular hypergraph
  • Mixed hypergraph
  • Transversal number
  • Upper chromatic number
  • Vertex coloring

ASJC Scopus subject areas

  • Geometry and Topology

Cite this