[Paper Review] Coloring Distance Graphs on the Integers

Glenn G. Chappell|ArXiv.org|May 19, 1998

Limits and Structures in Graph Theory9 references3 citations

TL;DR

This paper investigates the chromatic number of distance graphs defined on the integers, where edges connect vertices differing by a distance in a given set D. For sets D = {d₁, d₂, d₃, ...} with dₙ | dₙ₊₁, the authors prove such graphs admit a proper 4-coloring and determine their exact chromatic numbers, establishing a link between periodic colorings and chromatic number in these structures.

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.

Motivation & Objective

• To determine the chromatic number of distance graphs on the integers for sets D with dₙ | dₙ₊₁.
• To investigate the existence and structure of periodic proper colorings in such distance graphs.
• To establish a relationship between the chromatic number and the existence of periodic colorings.
• To characterize all distance graphs with such divisibility conditions that admit proper colorings.
• To provide exact values for the chromatic numbers of these specific distance graphs.

Proposed method

• Define a distance graph G(Z, D) with vertex set Z and edges between integers differing by a value in D.
• Use the divisibility condition dₙ | dₙ₊₁ to construct a periodic coloring scheme with period dividing dₙ for increasing n.
• Prove that a 4-coloring exists by constructing a coloring based on residue classes modulo 4, leveraging the divisibility structure.
• Analyze the structure of the graph using properties of integer lattices and divisibility chains.
• Show that the chromatic number is exactly 4 when D satisfies the divisibility condition.
• Establish a necessary and sufficient condition for the existence of periodic proper colorings in terms of the structure of D.

Experimental results

Research questions

• RQ1 What is the chromatic number of a distance graph on the integers when the distance set D satisfies dₙ | dₙ₊₁ for all n?
• RQ2 Under what conditions does a distance graph on the integers admit a periodic proper coloring?
• RQ3 How does the chromatic number relate to the existence of periodic colorings in such graphs?
• RQ4 Can the chromatic number be exactly determined for distance sets with the divisibility property?
• RQ5 What structural properties of D ensure that the chromatic number is bounded above by 4?

Key findings

• For any distance set D = {d₁, d₂, d₃, ...} with dₙ | dₙ₊₁ for all n, the associated distance graph has chromatic number exactly 4.
• A proper 4-coloring exists for all such distance graphs, constructed using residue classes modulo 4.
• The chromatic number is bounded above by 4, and this bound is tight for certain distance sets.
• Periodic proper colorings exist if and only if the distance set D satisfies the divisibility condition dₙ | dₙ₊₁.
• The existence of a periodic coloring implies that the chromatic number is finite and can be explicitly computed.
• The paper fully characterizes the chromatic number of distance graphs under the divisibility condition, showing it is always 4 or less.