Summary: Covering a hypergraph of subgraphs
Dedicated to Dan Kleitman, for his 65th-birthday.
Let G be a tree and let H be a collection of subgraphs of G, each having at most d connected
components. Let (H) denote the maximum number of members of H no two of which share
a common vertex, and let (H) denote the minimum cardinality of a set of vertices of G that
intersects all members of H. It is shown that (H) 2d2
(H). A similar, more general result
is proved replacing the assumption that G is a tree by the assumption that it has a bounded
tree-width. These improve and extend results of various researchers.
Let H be a finite collection of subgraphs of a finite graph G. The covering number (or piercing number)
(H) of H is the minimum cardinality of a set of vertices of G that intersects every member of H.
The matching number (H) of H is the maximum number of pairwise vertex disjoint members of H.
Clearly (H) (H). In general, (H) cannot be bounded from above by a function of (H), as
shown, for example, by all induced subgraphs on n vertices of an arbitrary graph on 2n - 1 vertices,
where = 1 and = n. If, however, the graph G is a tree and each member of H has at most d
connected components, then can be bounded by a function of and d.
Gallai noticed that if G is a path and d = 1 then = . More generally, Sur´anyi (see ) proved