On a theorem of lovász on covers inr-partite hypergraphs
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A theorem of Lovász asserts that τ(H)/τ*(H)≤r/2 for everyr-partite hypergraphH (where τ and τ* denote the covering number and fractional covering number respectively). Here it is shown that the same upper bound is valid for a more general class of hypergraphs: those which admit a partition (V1, ...,V k ) of the vertex set and a partitionp1+...+p k ofr such that |e⌢V i |≤p i ≤r/2 for every edgee and every 1≤i≤k. Moreover, strict inequality holds whenr>2, and in this form the bound is tight. The investigation of the ratio τ/τ* is extended to some other classes of hypergraphs, defined by conditions of similar flavour. Upper bounds on this ratio are obtained fork-colourable, stronglyk-colourable and (what we call)k-partitionable hypergraphs.
Mathematics Subject Classification (1991)05 C 65 05 D 15
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