Summary: On a hypergraph matching problem
Let H = (V, E) be an r-uniform hypergraph and let F 2V
. A matching M of H is (, F)-
perfect if for each F F, at least |F| vertices of F are covered by M. Our main result is
a theorem giving sufficient conditions for an r-uniform hypergraph to have a (1 - , F)-perfect
matching. As a special case of our theorem we obtain the following result. Let K(n, r) denote
the complete r-uniform hypergraph with n vertices. Let t and r be fixed positive integers where
t r 2. Then, K(n, r) can be packed with edge-disjoint copies of K(t, r) such that each
vertex is incident with only o(nr-1
) unpacked edges. This extends a result of Ršodl .
A hypergraph H is an ordered pair H = (V, E) where V is a finite set (the vertex set) and E is a
family of distinct subsets of V (the edge set). A hypergraph is r-uniform if all edges have size r.
In this paper we only consider r-uniform hypergraphs where r 2 is fixed. A subset M E(H) is
a matching if every pair of edges from M has an empty intersection. A matching is called perfect
if |M| = |V |/r. A vertex v V is covered by the matching M if some edge from M contains v.
Let F 2V and let 0 1. A matching M is (, F)-perfect if for each F F, at least |F|