Year: 2026
Type: preprint
Abstract: We study how the densities of different packing colors constrain one another in integer distance graphs. For D(1,6) we determine exact joint density regions and prove that the maximum combined density of colors 1 through 7 is 211/252. We determine the limiting color frequencies when this maximum is approached. Some optimal density vectors can be attained by nonperiodic colorings but by no periodic coloring; we quantify the deficit needed to switch between relevant configurations. For sufficiently large additional color indices, a fixed finite graph describes the joint density region. In the class 36+14t, t >= 0, we determine the full seven-vertex region and its earliest stable index. The proofs combine structural arguments with exact integer certificates. Applications give 17 <= chi_rho(D(1,6)) <= 20, 18 <= chi_rho(D(1,8)) <= 22, and chi_rho(D(1,9)) <= 17. Preprint. Not externally peer reviewed. The main paper is accompanied by a required technical supplement and computational evidence archives.
Source: arXiv (Cornell University)
Author: Zhang, Enkai
Institution: University of Toronto
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7211914072