Multipartite hypergraphs achieving equality in Ryser's conjecture

A famous conjecture of Ryser is that in an $r$-partite hypergraph the covering number is at most $r-1$ times the matching number. If true, this is known to be sharp for $r$ for which there exists a projective plane of order $r-1$. We show that the conjecture, if true, is also sharp for the smallest previously open value, namely $r=7$. For $r\in\{6,7\}$, we find the minimal number $f(r)$ of edges in an intersecting $r$-partite hypergraph that has covering number at least $r-1$. We find that $f(r)$ is achieved only by linear hypergraphs for $r\le5$, but that this is not the case for $r\in\{6,7\}$. We also improve the general lower bound on $f(r)$, showing that $f(r)\ge 3.052r+O(1)$. We show that a stronger form of Ryser's conjecture that was used to prove the $r=3$ case fails for all $r>3$. We also prove a fractional version of the following stronger form of Ryser's conjecture: in an $r$-partite hypergraph there exists a set $S$ of size at most $r-1$, contained either in one side of the hypergraph or in an edge, whose removal reduces the matching number by 1.Comment: Minor revisions after referee feedbac

The Disjoint Domination Game

We introduce and study a Maker-Breaker type game in which the issue is to create or avoid two disjoint dominating sets in graphs without isolated vertices. We prove that the maker has a winning strategy on all connected graphs if the game is started by the breaker. This implies the same in the $(2:1)$ biased game also in the maker-start game. It remains open to characterize the maker-win graphs in the maker-start non-biased game, and to analyze the $(a:b)$ biased game for $(a:b)\neq (2:1)$. For a more restricted variant of the non-biased game we prove that the maker can win on every graph without isolated vertices.Comment: 18 page