On the independence ratio of distance graphs

TL;DR

This paper investigates the independence ratio of distance graphs, proving it is achieved by periodic sets, and provides methods to determine exact or asymptotic ratios for various families of such graphs.

Contribution

It introduces a framework using discharging arguments to bound the independence ratio and determines exact ratios for multiple families of distance graphs.

Findings

Independence ratio equals the inverse of the fractional chromatic number.

The ratio is achieved by periodic independent sets.

Exact ratios are found for several families of distance graphs.

Abstract

A distance graph is an undirected graph on the integers where two integers are adjacent if their difference is in a prescribed distance set. The independence ratio of a distance graph $G$ is the maximum density of an independent set in $G$. Lih, Liu, and Zhu [Star extremal circulant graphs, SIAM J. Discrete Math. 12 (1999) 491--499] showed that the independence ratio is equal to the inverse of the fractional chromatic number, thus relating the concept to the well studied question of finding the chromatic number of distance graphs. We prove that the independence ratio of a distance graph is achieved by a periodic set, and we present a framework for discharging arguments to demonstrate upper bounds on the independence ratio. With these tools, we determine the exact independence ratio for several infinite families of distance sets of size three, determine asymptotic values for others, and present several conjectures.