Abstract
A well-known conjecture, often attributed to Ryser, states that the cover number of an r-partite r-uniform hypergraph is at most r−1 times larger than its matching number. Despite considerable effort, particularly in the intersecting case, this conjecture remains wide open, motivating the pursuit of variants of the original conjecture. Recently, Bustamante and Stein and, independently, Király and Tóthmérész considered the problem under the assumption that the hypergraph is t-intersecting, conjecturing that the cover number τ(H) of such a hypergraph H is at most r−t. In these papers, it was proven that the conjecture is true for r≤4t−1, but also that it need not be sharp; when r=5 and t=2, one has τ(H)≤2.We extend these results in two directions. First, for all t≥2 and r≤3t−1, we prove a tight upper bound on the cover number of these hypergraphs, showing that they in fact satisfy τ(H)≤⌊(r−t)/2⌋+1. Second, we extend the range of t for which the conjecture is known to be true, showing that it holds for all r≤367t−5. We also introduce several related variations on this theme. As a consequence of our tight bounds, we resolve the problem for k-wise t-intersecting hypergraphs, for all k≥3 and t≥1. We further give bounds on the cover numbers of strictly t-intersecting hypergraphs and the s-cover numbers of t-intersecting hypergraphs.
Highlights
We define an r-uniform hypergraph H to be r-partite if one can partition the vertex set into r parts, say V (H) = P1 · · · Pr, such that for all e ∈ E(H) and all j ∈ [r], we have |e∩Pj| = 1
Haxell and Scott [19] construct nearly-extremal intersecting hypergraphs; more precisely, they show Ryser(r, 1) ≥ r − 4 for all r large enough. This led Bustamante and Stein [6] and, independently, Király and Tóthmérész [23] to investigate what occurs when we impose the stricter condition of the hypergraph H being t-intersecting
Following Ryser’s Conjecture, we study how much smaller a cover one is guaranteed to find in r-uniform r-partite hypergraphs satisfying the more restrictive condition of being k-wise t-intersecting
Summary
The intersecting case comes from a connection with a conjecture of Gyárfás [17] which states that the vertices of any r-edge-coloured clique can be covered by at most r − 1 monochromatic trees This conjecture in the setting of coloured complete graphs is equivalent to Ryser’s conjecture for intersecting hypergraphs, see e.g. This led Bustamante and Stein [6] and, independently, Király and Tóthmérész [23] to investigate what occurs when we impose the stricter condition of the hypergraph H being t-intersecting In this case, any subset of r − t + 1 vertices from an edge must form a cover, and so we trivially have τ (H) ≤ r − t + 1. Ryser(r, t) ≥ r/t − 1 for many pairs (r, t), and Bustamante and Stein suggested this lower bound should be closer to the truth than the upper bound of Conjecture 1.2
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