Self-similarity of graphs

Abstract

An old problem raised independently by Jacobson and Schönheim 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 Erdős, Pach, and Pyber from 1987.

15 pages