Fundamental Groups of Hamming Graphs

TL;DR

This paper computes the fundamental groups of Hamming graphs, showing they are direct products of cyclic groups, and explores their implications for $ imes$-homotopy covers, advancing the understanding of graph homotopy theory.

Contribution

It provides the first complete computation of fundamental groups for all Hamming graphs and characterizes their structure as direct products of cyclic groups.

Findings

Fundamental groups of Hamming graphs are direct products of cyclic groups.

Explicit descriptions of some $ imes$-homotopy covers of Hamming graphs.

Enhanced understanding of graph homotopy and cover theory.

Abstract

Recently there has been growing interest in discrete homotopies and homotopies of graphs beyond treating graphs as 1-dimensional simplicial spaces. One such type of homotopy is $\times$-homotopy. Recent work by Chih-Scull has developed a homotopy category, a fundamental group for graphs under this homotopy, and a way of computing covers of graphs that lift homotopy via this fundamental group. In this paper, we compute the fundamental groups of all Hamming graphs, show that they are direct products of cyclic groups, and use this result to describe some $\times$-homotopy covers of Hamming graphs.