Abstract:A G-design of order n is a decomposition of the complete graph on n vertices into edge-disjoint subgraphs isomorphic to G. We survey the current state of knowledge on the existence problem for G-designs. This includes references to all the necessary designs and constructions, as well as a few new designs.

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It is shown that any partial Latin square of order at least six which consists of two filled rows and two filled columns can be completed.

We establish new lower bounds on the pair covering number C λ (v, k) for infinitely many values of v, k and λ, including infinitely many values of v and k for λ = 1. Here, C λ (v, k) denotes the minimum number of k-subsets of a v-set of points such that each pair of points occurs in at least λ of the k-subsets. We use these results to prove simple numerical conditions which are both necessary and sufficient for the existence of (K k − e)-designs with more points than blocks.