Random walks on quasirandom graphs

TL;DR

This paper proves that a random walk on a quasirandom graph of size n, with length proportional to n^2, results in a set of traversed edges that also form a quasirandom graph, and extends this to random embeddings of trees.

Contribution

It provides a positive answer to whether random walks on quasirandom graphs produce quasirandom edge sets and extends the result to random embeddings of trees.

Findings

Random walks of length alpha n^2 on quasirandom graphs produce quasirandom edge sets.

The result extends to random embeddings of trees.

Supports the robustness of quasirandom properties under certain random processes.

Abstract

Let G be a quasirandom graph on n vertices, and let W be a random walk on G of length alpha n^2. Must the set of edges traversed by W form a quasirandom graph? This question was asked by B\"ottcher, Hladk\'y, Piguet and Taraz. Our aim in this paper is to give a positive answer to this question. We also prove a similar result for random embeddings of trees.