A combinatorial approach to phase transitions in random graph isomorphism problems

We consider two independent Erdős-Rényi 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 Erdős-Rényi random graphs. Our approach is primarily combinatorial, naturally leading to several related problems for further exploration.