Covering a hypergraph of subgraphs Dedicated to Dan Kleitman, for his 65th-birthday. Summary: Covering a hypergraph of subgraphs Noga Alon Dedicated to Dan Kleitman, for his 65th-birthday. Abstract 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. 1 Introduction 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 [4]) proved Collections: Mathematics