Smallest set-Transversals of k-Partitions

Csilla Bujtás, Zsolt Tuza

Research output: Contribution to journalArticle

6 Citations (Scopus)

Abstract

We asymptotically solve an open problem raised independently by Sterboul (Colloq Math Soc J Bolyai 3:1387-1404, 1973), Arocha et al. (J Graph Theory 16:319-326, 1992) and Voloshin (Australas J Combin 11:25-45, 1995). For integers n ≥ k ≥ 2, let f(n, k) denote the minimum cardinality of a famil H of k-element sets over an n-element underlying set X such that every partition X1 ∪. . .∪ Xk = X into k nonempty classes completely partitions some H ∈ H; that is, {pipe}H ∩ Xi{pipe} = 1 holds for all 1 ≤ i ≤ k. This very natural function-whose defining property for k = 2 just means that H is a connected graph-turns out to be related to several extensively studied areas in combinatorics and graph theory. We prove general estimates from which, follows for every fixed k, and also for all k = o(n1/3), as n → ∞. Further, we disprove a conjecture of Arocha et al. (1992). The exact determination of f(n,k) for all n and k appears to be far beyond reach to our present knowledge, since e.g. the equality, holds, where the last term is the Turán number for graphs of girth 5.

Original languageEnglish
Pages (from-to)807-816
Number of pages10
JournalGraphs and Combinatorics
Volume25
Issue number6
DOIs
Publication statusPublished - Mar 1 2010

Keywords

  • C-coloring
  • Cochromatic number
  • Crossing set
  • Hypergraph
  • Partition
  • Upper chromatic number
  • Vertex coloring

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics

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