# The perfect 2-colorings of infinite circulant graphs with a continuous set of odd distances

## TL;DR

This paper classifies perfect 2-colorings of certain infinite circulant graphs with odd distances and proposes a conjecture for generalizing to more colors.

## Contribution

It provides a complete enumeration of perfect 2-colorings for these graphs and introduces a conjecture extending the results to any number of colors.

## Key findings

- Enumerated all perfect 2-colorings for the given graphs.
- Formulated a conjecture for perfect colorings with more than two colors.

## Abstract

A vertex coloring of a given simple graph $G=(V,E)$ with $k$ colors ($k$-coloring) is a map from its vertex set to the set of integers $\{1,2,3,\dots, k\}$. A coloring is called perfect if the multiset of colors appearing on the neighbours of any vertex depends only on the color of the vertex. We consider perfect colorings of Cayley graphs of the additive group of integers with generating set $\{1,-1,3,-3,5,-5,\dots, 2n-1,1-2n\}$ for a positive integer $n$. We enumerate perfect $2$-colorings of the graphs under consideration and state the conjecture generalizing the main result to an arbitrary number of colors.