2014/01/28 by James M. Carraher, David Galvin, Carraher, James M. +7 · 2 citations
Computer Science · Mathematics · #05C15 #05C35 #05C85 #Advanced Graph Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Graph theory and applications #Limits and Structures in Graph Theory #math.CO #msc:05C15 #msc:05C35 #msc:05C85
paper · pdf · doi:10.48550/arxiv.1401.7183
39 pages, 12 figures, 6 tables
arxiv created 2014/01/28 · openalex publication_date 2014/01/28 · arxiv updated 2014/01/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.