Simple versus nonsimple loops on random regular graphs

Abstract

In this note, we solve the “birthday problem” for loops on random regular graphs. Namely, for fixed , we prove that on a random ‐regular graph with vertices, as approaches infinity, with high probability: (i) almost all primitive nonbacktracking loops of length are simple, that is, do not self‐intersect, and (ii) almost all primitive nonbacktracking loops of length self‐intersect.