On a theorem of lovász on covers inr-partite hypergraphs

Abstract

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 (V 1, ...,V k ) of the vertex set and a partitionp 1+...+p k ofr such that |eV i |≤p i r/2 for every edgee and every 1≤ik. 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.

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Supported by grant HL28438 at MIPG, University of Pennsylvania, and by the fund for the promotion of research at the Technion.

This author's research was supported by the fund for the promotion of research at the Technion.

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Aharoni, R., Holzman, R. & Krivelevich, M. On a theorem of lovász on covers inr-partite hypergraphs. Combinatorica 16, 149–174 (1996). https://doi.org/10.1007/BF01844843

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Mathematics Subject Classification (1991)

  • 05 C 65
  • 05 D 15