Proportional graphs

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Abstract
Let S be a finite set of graphs and t a real number, 0 < t < 1. A (deterministic) graph G is (t, 5)‐proportional if for every H ∈ S, the number of induced subgraphs of G isomorphic to H equals the expected number of induced copies of H in the random graph Gn, t where n = |V(G)|. Let Sk = {all graphs on k vertices}, in particular S3 = {K3, P2, K2Kt, D3}. The notion of proportional graphs stems from the study of random graphs (Barbour, Karoński, and Ruciński, J Combinat. Th. Ser. B, 47, 125‐145, 1989; Janson and Nowicki, Prob. Th. Rel. Fields, to appear, Janson, Random Struct. Alg., 1, 15‐37, 1990) where it is shown that (t, S3)‐proportional graphs play a very special role; we thus call them simply t‐proportional. However, only a few ½‐proportional graphs on 8 vertices were known and it was an open problem whether there are any f‐proportional graphs with t ≠ ½ at all. In this paper, we show that there are infinitely many ½‐proportional graphs and that there are t‐proportional graphs with t≠. Both results are proved constructively. [We are not able to provide the latter construction for all f∈ Q∩(0,1), but the set of ts for which our construction works is dense in (0,1).] To support a conviction that the existence of (t, S3)‐proportional graphs was not quite obvious, we show that there are no (t, S4)‐proportional graphs.

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