Fractional v integral covers in hypergraphs of bounded edge size

[摘要] In the early 1980's, V. Rodl proved the Erdos-Hanani Conjecture, sparking a remarkable sequence of developments in the theory of packing and covering in hypergraphs of bounded edge size, Generalizations were given by P. Frankl and Rodl, by N. Pippenger, and by others. In each case, an appropriate semi-random method was used to ''construct'' the desired optimal object (covering, matching, colouring) in several random stages, followed by a greedy stage. The current work, which further generalizes some of the above results, is again probabilistic, and uses, in addition to earlier ideas, connections with so-called hard-core distributions on the set of matchings of a graph. For fixed k greater than or equal to 2, H a k-bounded hypergraph, and t: H --> R+ a fractional cover, a sufficient condition is given to ensure that the edge cover number p(H), i.e., the size of a smallest set of edges of H with union V(H), is asymptotically at most t(H) = Sigma(A is an element of H)t(A). This settles a conjecture first publicized in Visegrad, June 1991. (C) 1997 Academic Press.

[发布日期] 1997-05-01