Self-Similarity of Graphs

An old problem raised independently by Jacobson and Sch¨onheim asks to determine the maximum s for which every graph with m edges contains a pair of edge-disjoint isomorphic subgraphs with s edges. In this paper we determine this maximum up to a constant factor. We show that every m-edge graph contains a pair of edge-disjoint isomorphic subgraphs with at least c(m log m) 2/3 edges for some absolute constant c, and find graphs where this estimate is off only by a multiplicative constant. Our results improve bounds of Erdos, Pach, and Pyber from 1987.