  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.
