Let v≥k≥1 and λ≥0 be integers. Recall that a (v, k, λ) block design is a collection B of k-subsets of a v-set X in which every unordered pair of elements in X is contained in exactly λ of the subsets in B. Now let G be a graph with no multiple edges. A (v, G, λ) graph design is a collection H of subgraphs, each isomoprhic to G, of the complete graph K v such that each edge of K v appears in exactly λof the subgraphs in H. A famous result of Wilson states that for a fixed simple graph G and integer λ, there exists a (v, G, λ) graph design for all sufficiently large integers v satisfying certain necessary conditions. Here, we extend this result to include the case of loops in G. As a consequence, we obtain the asymptotic existence of equireplicate graph designs. Applications of the equireplicate condition are given.
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