Small subgraphs in the trace of a random walk

TL;DR

This paper investigates when fixed subgraphs appear in the trace of a random walk on complete and random graphs, revealing thresholds and differences in appearance times for various subgraph types.

Contribution

It establishes thresholds for subgraph appearance in random walk traces and compares these to thresholds in Erdős–Rényi graphs, highlighting differences for forests.

Findings

Subgraphs with cycles appear at similar thresholds in both models.

Forests appear earlier in random walk traces than in G(n,m).

Thresholds for subgraph appearance depend on the subgraph's structure.

Abstract

We consider the combinatorial properties of the trace of a random walk on the complete graph and on the random graph $G(n,p)$. In particular, we study the appearance of a fixed subgraph in the trace. We prove that for a subgraph containing a cycle, the threshold for its appearance in the trace of a random walk of length $m$ is essentially equal to the threshold for its appearance in the random graph drawn from $G(n,m)$. In the case where the base graph is the complete graph, we show that a fixed forest appears in the trace typically much earlier than it appears in $G(n,m)$.