Ryser's conjecture

 Importance: High ✭✭✭
 Author(s): Ryser, Herbert J.
 Subject: Graph Theory » Hypergraphs
 Keywords: hypergraph matching packing
 Posted by: mdevos on: March 19th, 2007

\begin{conjecture} Let $H$ be an $r$-\Def[uniform]{hypergraph} $r$-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 $\tau \le (r-1) \nu$. \end{conjecture}

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 $\tau \le \frac{1}{2} r \nu^*$.

Bibliography

[A] R. Aharoni, Ryser's conjecture for tripartite 3-graphs. Combinatorica 21 (2001), no. 1, 1--4. \MRhref{1805710}

[AH] R. Aharoni and P. Haxell, Hall's theorem for hypergraphs. J. Graph Theory 35 (2000), no. 2, 83--88. \MRhref{1781189}

[F] Z. Füredi, Maximum degree and fractional matchings in uniform hypergraphs, Combinatorica 1 (1981), 155--162. \MRhref{0625548}

[K] D. König, Theorie der endlichen und unendlichen Graphen, Leipzig, 1936.

[L] L. Lovász, On minimax theorems of combinatorics, Ph.D thesis, Matemathikai Lapok 26 (1975), 209--264 (in Hungarian). \MRhref{0510823}

* indicates original appearance(s) of problem.