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arXiv · 2608.26321

The Birthday Paradox for non-backtracking walks on regular graphs

Benjamin Dozier
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Abstract

We show a birthday paradox for random non-backtracking walk on regular graphs of degree at least $3$: such a walk of length $k$ has high probability of self-intersecting when $k$ is significantly greater than $\sqrt n$, where $n$ is the number of vertices of the graph. This resolves a conjecture of Noga Alon and Yuval Peres for the fixed degree case.

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Benjamin Dozier. 2026-08-26. The Birthday Paradox for non-backtracking walks on regular graphs. https://arxiv.org/abs/2608.26321

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