A famous conjecture of Tuza [12] 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.

An r-segment hypergraph H is a hypergraph whose edges consist of r consecutive integer points on line segments in R 2 . In this paper, we bound the chromatic number χ(H) and covering number τ (H) of hypergraphs in this family, uncovering several interesting geometric properties in the process. We conjecture that for r ≥ 3, the covering number τ (H) is at most (r − 1)ν(H), where ν(H) denotes the matching number of H. We prove our conjecture in the case where ν(H) = 1, and provide improved (in fact, optimal) bounds on τ (H) for r ≤ 5. We also provide sharp bounds on the chromatic number χ(H) in terms of r, and use them to prove two fractional versions of our conjecture. 1 arXiv:1807.04826v1 [math.CO]

The triangle packing number v(G) of a graph G is the maximum size of a set of edge-disjoint triangles in G. Tuza conjectured that in any graph G there exists a set of at most 2v(G) edges intersecting every triangle in G. We show that Tuza’s conjecture holds in the random graph G = G(n, m), when m ⩽ 0.2403n3/2 or m ⩾ 2.1243n3/2. This is done by analysing a greedy algorithm for finding large triangle packings in random graphs.

A d-interval is a union of at most d disjoint closed intervals on a fixed line. Tardos [9] and the second author [6] used topological tools to bound the transversal number τ of a family H of d-intervals in terms of d and the matching number ν of H. We investigate the weighted and fractional versions of this problem and prove upper bounds that are tight up to constant factors. We apply both the topological method and an approach of Alon [1]. For the use of the latter, we prove a weighted version of Turán's theorem. We also provide a proof of the upper bound of [6] that is more direct than the original proof.

Consider n players having preferences over the connected pieces of a cake, identified with the interval [0, 1]. A classical theorem, found independently by Stromquist and by Woodall in 1980, ensures that, under mild conditions, it is possible to divide the cake into n connected pieces and assign these pieces to the players in an envy-free manner, i.e, such that no player strictly prefers a piece that has not been assigned to her. One of these conditions, considered as crucial, is that no player is happy with an empty piece. We prove that, even if this condition is not satisfied, it is still possible to get such a division when n is a prime number or is equal to 4. When n is at most 3, this has been previously proved by Erel Segal-Halevi, who conjectured that the result holds for any n. The main step in our proof is a new combinatorial lemma in topology, close to a conjecture by Segal-Halevi and which is reminiscent of the celebrated Sperner lemma: instead of restricting the labels that can appear on each face of the simplex, the lemma considers labelings that enjoy a certain symmetry on the boundary.