Coloring Distance Graphs on the Integers

TL;DR

This paper studies the coloring of distance graphs on integers, proving that certain divisibility-based graphs are 4-colorable and determining their exact chromatic numbers, while also exploring periodic coloring properties.

Contribution

It characterizes the chromatic numbers of divisibility-based distance graphs and links these to the existence of periodic proper colorings.

Findings

Distance graphs with divisibility conditions are 4-colorable.

Exact chromatic numbers are determined for these graphs.

A relationship between chromatic number and periodic colorings is established.

Abstract

Given a set D of positive integers, the associated distance graph on the integers is the graph with the integers as vertices and an edge between distinct vertices if their difference lies in D. We investigate the chromatic numbers of distance graphs. We show that, if $D = {d_1,d_2,d_3,...}$, with $d_n | d_{n+1}$ for all n, then the distance graph has a proper 4-coloring. We further find the exact chromatic numbers of all such distance graphs. Next, we characterize those distance graphs that have periodic proper colorings and show a relationship between the chromatic number and the existence of periodic proper colorings.