Resumen de Asymptotic Existence of Resolvable Graph Designs

Peter Dukes, Alan C. H. Ling

• Let v \ge k \ge 1 and lambda \ge 0 be integers. A block design BD(v,k,lambda) is a collection cal A of k-subsets of a v-set X in which every unordered pair of elements from X is contained in exactly lambda elements of cal A. More generally, for a fixed simple graph G, a graph design GD(v,G,lambda) is a collection cal A of graphs isomorphic to G with vertices in X such that every unordered pair of elements from X is an edge of exactly $lambda$ elements of cal A. A famous result of Wilson says that for a fixed G and lambda, there exists a GD(v,G,lambda) for all sufficiently large v satisfying certain necessary conditions. A block (graph) design as above is resolvable if cal A can be partitioned into partitions of (graphs whose vertex sets partition) X. Lu has shown asymptotic existence in v of resolvable BD(v,k,lambda), yet for over twenty years the analogous problem for resolvable GD(v,G,lambda) has remained open. In this paper, we settle asymptotic existence of resolvable graph designs