Abstract
The symmetric difference of two graphs G1,G2 on the same set of vertices [n]={1,2,…,n} is the graph on [n] whose set of edges are all edges that belong to exactly one of the two graphs G1,G2. Let H be a fixed graph with an even (positive) number of edges, and let DH(n) denote the maximum possible cardinality of a family of graphs on [n] containing no two members whose symmetric difference is a copy of H. Is it true that DH(n)=o(2n2) for any such H? We discuss this problem, compute the value of DH(n) up to a constant factor for stars and matchings, and discuss several variants of the problem including ones that have been considered in earlier work.
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