Generalizations and Strengthenings of Ryser's Conjecture

Abstract

Ryser's conjecture says that for every $r$-partite hypergraph $H$ with matching number $\nu(H)$, the vertex cover number is at most $(r-1)\nu(H)$.  This far-reaching generalization of König's theorem is only known to be true for $r\leq 3$, or when $\nu(H)=1$ and $r\leq 5$.  An equivalent formulation of Ryser's conjecture is that in every $r$-edge coloring of a graph $G$ with independence number $\alpha(G)$, there exists at most $(r-1)\alpha(G)$ monochromatic connected subgraphs which cover the vertex set of $G$.   We make the case that this latter formulation of Ryser's conjecture naturally leads to a variety of stronger conjectures and generalizations to hypergraphs and multipartite graphs.  Regarding these generalizations and strengthenings, we survey the known results, improving upon some, and we introduce a collection of new problems and results.