Kőnig's theorem states that the covering number and the matching number of a bipartite graph are equal. We prove a generalization, in which the point in one fixed side of the graph of each edge is replaced by a subtree of a given tree. The proof uses a recent extension of Hall's theorem to families of hypergraphs, by the first author and P. Haxell [2]. As an application we prove a special case (that of chordal graphs) of a conjecture of B. Reed.
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Received January 27, 2000/Revised November 2, 2000
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ID=" " The research of the first author was supported by grants from the Israel Science Foundation, the M. & M.L Bank Mathematics Research Fund and the fund for the promotion of research at the Technion.
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Aharoni, R., Berger, E. & Ziv, R. A Tree Version of Kőnig's Theorem. Combinatorica 22, 335–343 (2002). https://doi.org/10.1007/s004930200016
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DOI: https://doi.org/10.1007/s004930200016