SELF-SIMILARITY OF GRAPHS.

An old problem raised independently by Jacobson and Schönheim seeks to determine the maximum for which every graph with 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 Erdös, Pach, and Pyber from 1987. [ABSTRACT FROM AUTHOR]