Near Perfect Coverings in Graphs and Hypergraphs

Abstract

Suppose we are given a bipartite graph with vertex set X, Y, |X| = n , |Y| = N , each point in X (Y) has degree D(d) fixed, respectively, moreover, each pair of points x, x' ∈ X has at most D/(log n) 3 (common) neighbours. Let t (X, Y) denote the minimum number of vertices of Y needed to cover all vertices of X. We prove (Theorem 1.1) that t (X, Y) d / n tends to 1 as n tends to infinity. This result has many applications: Theorem [5]. Suppose k > r > 1 are fixed, n → ∞. Then there exists a collection of (1 + o (1)) × ( n r )/( k r ) k -subsets of an n -set so that each r -subset is contained in at least one member of the collection. Analogues and strengthenings of this result are deduced. E.g. for vector spaces, orthogonal or simplectic geometries, random collections of k -sets with constant probabilities, etc. Theorem 3.3. Suppose G is a graph on v vertices and e edges and ℛ is a random graph on n vertices and edge probability e / ( v 2 ). Then there exists a collection of (1 + o (1))( n 2 )/ ( v 2 ) induced subgraphs of ℛ on v vertices, isomorphic to G and such that each edge (non-edge) of ℛ is covered by an edge (non-edge) of a graph in the collection.