On the number of circuits in random graphs

We apply in this article (non rigorous) statistical mechanics methods to the problem of counting long circuits in graphs. The outcomes of this approach have two complementary flavours. On the algorithmic side, we propose an approximate counting procedure, valid in principle for a large class of graphs. On a more theoretical side, we study the typical number of long circuits in random graph ensembles, reproducing rigorously known results and stating new conjectures.