Random Walk Labelings of Perfect Trees and Other Graphs

TL;DR

This paper computes the number of possible random walk labelings for various classes of graphs, including perfect trees, combs, double combs, tori, and certain cycle-connected graphs, extending recent research in the area.

Contribution

It provides explicit calculations of random walk labelings for new classes of graphs, advancing understanding of graph labelings derived from random walks.

Findings

Number of labelings for perfect trees, combs, and double combs

Labelings for torus $C_2\times C_n$

Labelings for graphs formed by connecting three paths into cycles

Abstract

A Random walk labeling of a graph $G$ is any labeling of $G$ that could have been obtained by performing a random walk on $G$. Continuing two recent works, we calculate the number of random walk labelings of perfect trees, combs, and double combs, the torus $C_2\times C_n$, and the graph obtained by connecting three path graphs to form two cycles.