On a theorem of Lovász on covers in r-partite hypergraphs

A theorem of Lovász asserts that τ(H)/τ* (H)≤r/2 for every r-partite hypergraph H (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 partition p₁ +...+ p_k of r such that |e∩V_i|≤p_i≤r/2 for every edge e and every 1≤i≤k. Moreover, strict inequality holds when r>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 for k-colourable, strongly k-colourable and (what we call) k-partitionable hypergraphs.