Open Problem Garden

Importance: High ✭✭✭

Author(s):

Subject:	Graph Theory
	» Hypergraphs

Keywords:	hypergraph
	matching
	packing

Recomm. for undergrads: no

Posted	by:	mdevos
	on:	March 19th, 2007

Conjecture Let $H$ be an $r$ -uniform $r$ -partite hypergraph. If $\nu$ is the maximum number of pairwise disjoint edges in $H$ , and $\tau$ is the size of the smallest set of vertices which meets every edge, then $\tau \le (r-1) \nu$ .

Definitions: A (vertex) cover is a set of vertices which meets (has nonempty intersection with) every edge, and we let $\tau(H)$ denote the size of the smallest vertex cover of $H$ . A matching is a collection of pairwise disjoint edges, and we let $\nu(H)$ denote the size of the largest matching in $H$ . When the hypergraph is clear from context, we just write $\tau$ or $\nu$ .

It is immediate that $\nu \le \tau$ , since every cover must contain at least one point from each edge in any matching. For $r$ -uniform hypergraphs, $\tau \le r \nu$ , since the union of the edges from any maximal matching is a set of at most $r \nu$ vertices that which meets every edge. Ryser's conjecture is that this second bound can be improved if $H$ is $r$ -uniform and $r$ -partite (the vertices may be partitioned into $r$ sets $V_1,V_2,\ldots,V_r$ so that every edge contains exactly one element of each $V_i$ ).

In the special case when $r=2$ our trivial inequality yields $\nu \le \tau$ and the conjecture implies $\tau \le \nu$ , so we should have $\nu = \tau$ . In fact this is true, it is König's theorem on bipartite graphs [K]. Indeed, Ryser's conjecture is probably easiest to view as a high dimensional generalization of this early result of König. Recently, Aharoni [A] has applied the "Hall's theorem for hypergraphs" result of Aharoni and Haxell [AH] to prove this conjecture for $r=3$ . However the case $r=4$ is still wide open.

Some other interesting work on this problem concerns fractional covers and fractional matchings. A fractional cover of $H = (V,E)$ is a weighting $a : V \rightarrow {\mathbb R}^+$ so that $\sum_{x \in S} a(x) \ge 1$ for every $S \in E$ , and the weight of this cover is $\sum_{x \in V} a(x)$ . The fractional cover number, denoted $\tau^*$ is the infimum of the set of weights of covers. Similarly, a fractional matching is an edge-weighting $b : E \rightarrow {\mathbb R}^+$ so that $\sum_{S \ni x} b(S) \le 1$ for every $x \in V$ , and the weight of this matching is $\sum_{S \in E} b(S)$ . The fractional matching number, denoted $\nu^*$ is the supremum of the set of weights of fractional matchings. Fractional covers and matchings are the usual fractional relaxations, and by LP-duality, they satisfy $\nu^* = \tau^*$ for every hypergraph. For $r$ -regular $r$ -partite hypergraphs, Füredi [F] has proved that $\tau^* \le (r-1)\nu$ and Lovasz [L] has shown $\tau \le \frac{1}{2} r \nu^*$ .