A Lower Bound on the Area of a 3-Coloured Disk Packing

TL;DR

This paper investigates the minimum area coverage achievable by a 3-colourable subset of disks in the plane, providing bounds, conjectures, and an efficient algorithm related to wireless channel assignment.

Contribution

It introduces a novel variant of disk packing constrained by 3-colorability, offers new bounds and conjectures, and presents an efficient algorithm for subset selection.

Findings

Proved a lower bound of 1/2.09 for the area coverage.

Conjectured a coverage of at least 1/1.41 of the union area.

Developed an O(n^2) algorithm achieving a 1/2.77 coverage bound.

Abstract

Given a set of unit-disks in the plane with union area $A$, what fraction of $A$ can be covered by selecting a pairwise disjoint subset of the disks? Rado conjectured 1/4 and proved $1/4.41$. Motivated by the problem of channel-assignment for wireless access points, in which use of 3 channels is a standard practice, we consider a variant where the selected subset of disks must be 3-colourable with disks of the same colour pairwise-disjoint. For this variant of the problem, we conjecture that it is always possible to cover at least $1/1.41$ of the union area and prove $1/2.09$. We also provide an $O(n^2)$ algorithm to select a subset achieving a $1/2.77$ bound.