GENERALIZED TUZA'S CONJECTURE FOR RANDOM HYPERGRAPHS

A celebrated conjecture of Tuza states that in any finite graph the minimum size of a cover of triangles by edges is at most twice the maximum size of a set of edge-disjoint triangles. For an r-uniform hypergraph (r-graph) G , let \tau(G) be the minimum size of a cover of edges by (r- 1)- sets of vertices, and let v ( G ) be the maximum size of a set of edges pairwise intersecting in fewer than r- 1 vertices. Aharoni and Zerbib proposed the following generalization of Tuza's conjecture: For any r-graph G , \tau(G)/v(G) \leq [(r + 1)/21. Let Hr(n,p) be the uniformly random r-graph on n vertices. We show that for r \in {3, 4, 5\} and any p = p ( n ), Hr(n, p ) satisfies the Aharoni-Zerbib conjecture with high probability (w.h.p.), i.e., with probability approaching 1 as n- oo. We also show that there is a C < 1 such that for any r \geq 6 and any p = p ( n ), \tau(Hr(n,p))/v(Hr(n,p)) \leq Cr w.h.p. Furthermore, we may take C < 1/2 + \varepsilon, for any \varepsilon > 0, by restricting to sufficiently large r (depending on \varepsilon).