Simple vs non-simple loops on random regular graphs

TL;DR

This paper investigates the properties of loops in random regular graphs, showing a phase transition at length proportional to the square root of the number of vertices, distinguishing simple from self-intersecting loops.

Contribution

It provides a rigorous analysis of the transition point for loop simplicity in random regular graphs, solving a version of the birthday problem for these structures.

Findings

Loops of length less than sqrt(n) are mostly simple.

Loops of length greater than sqrt(n) mostly self-intersect.

Identifies a phase transition at loop length proportional to sqrt(n).

Abstract

In this note we solve the ``birthday problem'' for loops on random regular graphs. Namely, for fixed $d\ge 3$, we prove that on a random $d$-regular graph with $n$ vertices, as $n$ approaches infinity, with high probability: (i) almost all primitive non-backtracking loops of length $k \prec \sqrt{n}$ are simple, i.e. do not self-intersect, (ii) almost all primitive non-backtracking loops of length $k \succ \sqrt{n}$ self-intersect.