A limit theorem for the six-length of random functional graphs with a fixed degree sequence

Abstract

We obtain results on the limiting distribution of the six-length of a random functional graph, also called a functional digraph or random mapping, with given in-degree sequence. The six-length of a vertex $v\in V$ is defined from the associated mapping, $f:V\to V$, to be the maximum $i\in V$ such that the elements $v,f(v),\dots ,f^{i-1}(v)$ are all distinct. This has relevance to the study of algorithms for integer factorisation.