Small Edge Sets Meeting all Triangles of a Graph

S. Aparna Lakshmanan, Cs Bujtás, Zs Tuza

Research output: Contribution to journalArticle

10 Citations (Scopus)

Abstract

It was conjectured in 1981 by the third author that if a graph G does not contain more than t pairwise edge-disjoint triangles, then there exists a set of at most 2t edges that shares an edge with each triangle of G. In this paper, we prove this conjecture for odd-wheel-free graphs and for 'triangle-3-colorable' graphs, where the latter property means that the edges of the graph can be colored with three colors in such a way that each triangle receives three distinct colors on its edges. Among the consequences we obtain that the conjecture holds for every graph with chromatic number at most four. Also, two subclasses of K 4-free graphs are identified, in which the maximum number of pairwise edge-disjoint triangles is equal to the minimum number of edges covering all triangles. In addition, we prove that the recognition problem of triangle-3-colorable graphs is intractable.

Original languageEnglish
Pages (from-to)381-392
Number of pages12
JournalGraphs and Combinatorics
Volume28
Issue number3
DOIs
Publication statusPublished - May 1 2012

Keywords

  • Odd-wheel-free graphs
  • T-transversal
  • Triangle-3-colorable graphs

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics

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