Endomorphisms of The Hamming Graph and Related Graphs

TL;DR

This paper characterizes all singular endomorphisms of the Hamming graph and related graphs, showing they are induced by Latin hypercubes and extending results to various generalizations.

Contribution

It provides a complete description of singular endomorphisms for Hamming graphs and their variants, linking them to Latin hypercubes and generalizing to non-uniform vertex sets.

Findings

Singular endomorphisms are uniform and induced by Latin hypercubes.

Results extend to complements and graphs with different Hamming distance constraints.

Characterization applies to vertices in product groups with varying sizes.

Abstract

In this paper we determine all singular endomorphisms of the Hamming graph and other related graphs. The Hamming graph has vertices $\mathbb{Z}^{m}_n$ where two vertices are adjacent, if their Hamming distance is $1$. We show that its singular endomorphisms are uniform (each kernel has the same size) and that they are induced by Latin hypercubes (which essentially determines the number of singular endomorphisms). However, we do the same for its complement and some related graphs where the Hamming distance is allowed to be one of $1,...,k$, for some $1\leq k\leq m-1$. Ultimately, we consider the same situation where the vertices are tuples in $\mathbb{Z}_{n_1}\times\mathbb{Z}_{n_2}\times\cdots \times\mathbb{Z}_{n_m}$ (not all $n_i$ are equal).