The Numbers of Spanning Trees, Hamilton Cycles and Perfect Matchings in a Random Graph

Abstract

The numbers of spanning trees, Hamilton cycles and perfect matchings in a random graph Gnm are shown to be asymptotically normal if m is neither too large nor too small. At the lowest limit m ≍ n3/2, these numbers are asymptotically log-normal. For Gnp, the numbers are asymptotically log-normal for a wide range of p, including p constant. The same results are obtained for random directed graphs and bipartite graphs. The results are proved using decomposition and projection methods.