Asymptotics of Hypergraph Matching, Covering and Coloring Problems

A hypergraph H is simply a collection of subsets of a finite set, which we will always denote by V. Elements of V are called vertices and elements of H edges. A hypergraph is k-uniform (k-bounded) if each of its edges has size k (at most k). The degree in H of a vertex x is the number of edges containing x, and is denoted by d(x). Similarly, d(x,y) denotes the number of edges containing both of the vertices x, y. A hypergraph is d-regular if each of its vertices has degree d and simple if d(x,y) < 1 for all x, y. We use D(H) and 6(H) for the maximum and minimum degrees of H. A simple 2-uniform hypergraph is a graph, usually denoted G, whereas a general 2-uniform hypergraph is a multigraph. Principal objects associated with a hypergraph are matchings, covers, and colorings. A matching is a collection of pairwise disjoint edges. We write M. = M(H) for the set of matchings of H and v(H) for the matching number, the maximum size of a matching in H. A cover is a collection of edges whose union is V, and an (edge-)coloring is a : W —• S (S a set) with A D B ^ 0 => a(A) ^ a(B) (so a partition of H into matchings). For these the parameters analogous to v are p(H), the minimum size of a cover, and x'(^)> the minimum number of matchings in a coloring. For further background see e.g. [16]. Much of this talk deals with hypergraphs in which edge sizes are fixed or bounded and degrees arc large. In contrast to the familiar intractability of hypergraph problems, a central message here is that under the restrictions just stated one does often have, or at least seems to have, good asymptotic behavior. This is tied to notions of approximate independence (Sections 3., 4.) and relations between integer and linear programs (Section 5.). For a somewhat less compressed account of most of what is covered here, see [29] ; as discussed there (and apparent here), much of this material had its beginnings in problems of Paul Erdös.