Self-similarity of graphs

TL;DR

This paper determines the maximum size of edge-disjoint isomorphic subgraphs guaranteed in any graph with m edges, improving previous bounds and showing it scales roughly as (m log m)^{2/3}.

Contribution

It establishes tight bounds on the size of the largest pair of edge-disjoint isomorphic subgraphs in graphs with m edges, advancing understanding of graph self-similarity.

Findings

Every m-edge graph contains a pair of edge-disjoint isomorphic subgraphs with at least c (m log m)^{2/3} edges.

Constructed graphs show this bound is tight up to a constant factor.

Improves bounds previously established by Erdős, Pach, and Pyber in 1987.

Abstract

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 Erd\H{o}s, Pach, and Pyber from 1987.