A LOWER BOUND ON THE AREA OF A 3-COLOURED DISK PACKING

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(n2) algorithm to select a subset achieving a 1/2.77 bound. Finally, we discuss some results for other numbers of colours.