On the Last New Vertex Visited by a Random Walk in a Directed Graph

TL;DR

This paper proves that only directed cycles and complete graphs possess the property that a random walk starting at any vertex has an equally likely chance to end at any other vertex after visiting all vertices.

Contribution

It extends the characterization of graphs with uniform endpoint distribution in random cover tours to directed graphs, confirming only cycles and complete graphs have this property.

Findings

Only directed cycles and complete graphs have the property.

The result generalizes previous undirected graph findings.

Provides a complete classification for directed graphs.

Abstract

Consider a simple graph in which a random walk begins at a given vertex. It moves at each step with equal probability to any neighbor of its current vertex, and ends when it has visited every vertex. We call such a random walk a random cover tour. It is well known that cycles and complete graphs have the property that a random cover tour starting at any vertex is equally likely to end at any other vertex. Ronald Graham asked whether there are any other graphs with this property. In 1993, L\'aszlo Lov\'asz and Peter Winkler showed that cycles and complete graphs are the only undirected graphs with this property. We strengthen this result by showing that cycles and complete graphs (with all edges considered bidirected) are the only directed graphs with this property.