On a problem concerning integer distance graphs

TL;DR

This paper investigates the chromatic and clique numbers of integer distance graphs, disproving a conjecture that equates maximum clique size with chromatic number in certain cases.

Contribution

It provides a counterexample to a conjecture relating chromatic number and clique number in integer distance graphs, showing they can differ significantly.

Findings

Counterexamples with hi(G(D))=|D|+1 but irc(G(D))=|D|-1

Disproves the conjecture linking chromatic and clique numbers in this context

Establishes an infinite class of graphs with these properties

Abstract

For $D$ being a subset of positive integers, the integer distance graph is the graph $G(D)$, whose vertex set is the set of integers, and edge set is the set of all pairs $uv$ with $|u-v| \in D$. It is known that $\chi(G(D)) \leq |D|+1$. This article studies the problem (which is motivated by a conjecture of Zhu): "Is it true that $\chi(G(D)) = |D|+1$ implies $\omega(G(D)) \geq |D|+1$, where $\omega(H)$ is the clique number of $H$?". We give a negative answer to this question, by showing an infinite class of integer distance graphs with $\chi(G(D))=|D|+1$ but $\omega(G(D))=|D|-1$.