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 |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.