Generalizations and strengthenings of Ryser's conjecture

TL;DR

This paper explores generalizations and strengthenings of Ryser's conjecture, surveying known results, proposing new conjectures, and presenting related problems in hypergraph and multipartite graph theory.

Contribution

It introduces new conjectures and results that extend Ryser's conjecture, along with a comprehensive survey and analysis of existing work.

Findings

Survey of known results on Ryser's conjecture

Introduction of new conjectures and problems

Improved bounds and partial results for special cases

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 K\"onig's theorem is only known to be true for $r\leq 3$, or $\nu(G)=1$ 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.