A combinatorial approach to phase transitions in random graph isomorphism problems

TL;DR

This paper investigates phase transitions in the isomorphism of two independent Erdős-Rényi random graphs, revealing sharp asymptotic changes under certain parameter conditions and extending previous uniform graph results.

Contribution

It introduces a combinatorial approach to analyze phase transitions in non-uniform Erdős-Rényi graphs for isomorphism problems, expanding existing theoretical understanding.

Findings

Identifies sharp phase transition thresholds for graph isomorphism problems.

Extends known results from uniform to non-uniform Erdős-Rényi graphs.

Provides a combinatorial framework for future related problems.

Abstract

We consider two independent Erd\H{o}s-R\'enyi random graphs, with possibly different parameters, and study two isomorphism problems, a graph embedding problem and a common subgraph problem. Under certain conditions on the graph parameters we show a sharp asymptotic phase transition as the graph sizes tend to infinity. This extends known results for the case of uniform Erd\H{o}s-R\'enyi random graphs. Our approach is primarily combinatorial, naturally leading to several related problems for further exploration.