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Fundamental Groups of Hamming Graphs

Keira Behal, Tien Chih

TL;DR

The work develops a systematic computation of the $ imes$-homotopy fundamental groups for Hamming graphs $H(d,q)$. Using pleat structure, dimension-reduction of closed walks, and a set of commuting ground-walk generators $U_i$, the authors obtain explicit abelian group structures: $\\Pi(H(d,2))$ is trivial, $\\Pi(H(d,3)) \\cong \\mathbb{Z}^d$, and $\\Pi(H(d,q)) \\cong \\mathbb{Z}_2^d$ for $q>3$, together with a characterization of covers via these generators and the universal cover behavior. The results imply that Hamming graphs are cartesian products of complete graphs and establish connections to homotopy covers, with broader implications for product graphs $\\Pi(G \\\\square H) \\cong \\Pi(G) \\times \\Pi(H)$. The paper provides concrete descriptions of covers and universal covers, enriching the discrete homotopy theory of graphs under $\times$-homotopy.

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.

Fundamental Groups of Hamming Graphs

TL;DR

The work develops a systematic computation of the -homotopy fundamental groups for Hamming graphs . Using pleat structure, dimension-reduction of closed walks, and a set of commuting ground-walk generators , the authors obtain explicit abelian group structures: is trivial, , and for , together with a characterization of covers via these generators and the universal cover behavior. The results imply that Hamming graphs are cartesian products of complete graphs and establish connections to homotopy covers, with broader implications for product graphs . The paper provides concrete descriptions of covers and universal covers, enriching the discrete homotopy theory of graphs under -homotopy.

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 -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 -homotopy covers of Hamming graphs.
Paper Structure (6 sections, 16 theorems, 13 equations, 1 figure)

This paper contains 6 sections, 16 theorems, 13 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Hamming graph $H(d,2)$.
  4. General Hamming graphs $H(d,q)$, where $q\neq 2$.
  5. Conclusions and Future Directions
  6. Acknowledgments

Key Result

Proposition 2.23

Any connected homotopy cover $\widetilde{G}$ is of the form $U/S \cong \widetilde{G}$ where $S \leq \Pi(G)$ is the subgroup $\{s | s\tilde{\rho} = \tilde{\rho}\}$.

Figures (1)

  • Figure 1: The Hamming graphs $H(2,3)$.

Theorems & Definitions (65)

  • Definition 2.1: BMAGT
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4: CS1
  • Definition 2.5: CS2
  • Definition 2.6
  • Example 2.7
  • Definition 2.8: CS1
  • Definition 2.9: CS1
  • Remark 2.10
  • ...and 55 more