A generalization of Tuza's conjecture. Issue 3 (17th December 2019)

• Record Type: Journal Article
• Authors: Aharoni, Ron; Zerbib, Shira
• Is Part Of: Journal of graph theory. Volume 94:Issue 3(2020)
• Page Start: 445
• Page End: 462
• Publication Date: 2019-12-17
• Subjects: 3‐uniform hypergraphs -- covers -- matchings -- Tuza's conjecture; Graph theory -- Periodicals; 511
• DOI: 10.1002/jgt.22533
• Languages: English
• ISSNs: 0364-9024

Abstract

Abstract: A famous conjecture of Tuza is that the minimal number of edges needed to cover all triangles in a graph is at most twice the maximal size of a set of edge‐disjoint triangles. We propose a wider setting for this conjecture. For a hypergraph H let ν ( m ) ( H ) be the maximal size of a collection of edges, no two of which share m or more vertices, and let τ ( m ) ( H ) be the minimal size of a collection C of sets of m vertices, such that every edge in H contains a set from C . We conjecture that the maximal ratio τ ( m ) ( H ) / ν ( m ) ( H ) is attained in hypergraphs for which ν ( m ) ( H ) = 1 . This would imply, in particular, the following generalization of Tuza's conjecture: if H is 3‐uniform, then τ ( 2 ) ( H ) / ν ( 2 ) ( H ) ≤ 2 . (Tuza's conjecture is the case in which H is the set of all triples of vertices of triangles in any given graph.) We show that most known results on Tuza's conjecture go over to this more general setting. We also prove some general results on the ratio τ ( m ) ( H ) / ν ( m ) ( H ), and study the fractional versions and the case of k ‐partite hypergraphs.