Completely regular codes in the infinite hexagonal grid

TL;DR

This paper characterizes completely regular codes within the infinite hexagonal grid graph, providing a detailed understanding of their structure and properties in this specific geometric setting.

Contribution

It offers a complete characterization of completely regular codes in the infinite hexagonal grid, a novel contribution to the study of regular codes in geometric graphs.

Findings

Identifies all completely regular codes in the infinite hexagonal grid.

Provides explicit structural descriptions of these codes.

Enhances understanding of regularity conditions in geometric graph settings.

Abstract

A set $C$ of vertices of a simple graph is called a completely regular code if for each $i=0$, $1$, $2$, \ldots and $j = i-1$, $i$, $i+1$, all vertices at distance $i$ from $C$ have the same number $s_{ij}$ of neighbors at distance $j$ from $C$. We characterize the completely regular codes in the infinite hexagonal grid graph.