ON A THEOREM OF LOV ASZ ON COVERS IN r-PARTITE HYPERGRAPHS

A theorem of Lovv asz asserts that (H)== (H) r=2 for every r-partite hy-pergraph H (where and 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 partition p 1 + +p k of r such that je\V i j 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 == is extended to some other classes of hypergraphs, deened by conditions of similar avour. Upper bounds on this ratio are obtained for k-colourable, strongly k-colourable and (what we call) k-partitionable hypergraphs.