On Ryser's Conjecture for $t$-Intersecting and Degree-Bounded Hypergraphs

A famous conjecture (usually called Ryser’s conjecture) that appeared in the PhD thesis of his student, J. R. Henderson, states that for an r-uniform r-partite hypergraph H, the inequality τ(H) 6 (r − 1)·ν(H) always holds. This conjecture is widely open, except in the case of r = 2, when it is equivalent to Kőnig’s theorem, and in the case of r = 3, which was proved by Aharoni in 2001. Here we study some special cases of Ryser’s conjecture. First of all, the most studied special case is when H is intersecting. Even for this special case, not too much is known: this conjecture is proved only for r 6 5 by Gyárfás and Tuza. For r > 5 it is also widely open. Generalizing the conjecture for intersecting hypergraphs, we conjecture the following. If an r-uniform r-partite hypergraph H is t-intersecting (i.e., every two hyperedges meet in at least t < r vertices), then τ(H) 6 r − t. We prove this conjecture for the case t > r/4. Gyárfás showed that Ryser’s conjecture for intersecting hypergraphs is equivalent to saying that the vertices of an r-edge-colored complete graph can be covered by r − 1 monochromatic components. Motivated by this formulation, we examine what fraction of the vertices can be covered by r−1 monochromatic components of different colors in an r-edge-colored complete graph. We prove a sharp bound for this problem. Finally we prove Ryser’s conjecture for the very special case when the maximum degree of the hypergraph is two. ∗Research supported by a grant (no. K 109240) from the National Development Agency of Hungary, based on a source from the Research and Technology Innovation Fund. †Research was finished when the author was a visiting research fellow at Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences. the electronic journal of combinatorics 24(4) (2017), #P4.40 1