Covering a hypergraph of subgraphs

Authors:
Noga Alon
Affiliations:
Department of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Ramat Aviv, Israel
Venue:
Discrete Mathematics - Kleitman and combinatorics: a celebration
Year:
2002

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Abstract

Let G be a tree and let H be a collection of subgraphs of G, each having at most d connected components. Let v(H) denote the maximum number of members of H no two of which share a common vertex, and let τ(H) denote the minimum cardinality of a set of vertices of G that intersects all members of H. It is shown that τ(H) ≤ 2d2v(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.