 
Summary: ConflictFree colorings of Shallow Discs
Noga Alon
Shakhar Smorodinsky
Abstract
We prove that any collection of n discs in which each one intersects at most k others, can be
colored with at most O(log3
k) colors so that for each point p in the union of all discs there is at
least one disc in the collection containing p whose color differs from that of all other members
of the collection that contain p. This is motivated by a problem on frequency assignments in
cellular networks, and improves the best previously known upper bound of O(log n) when k is
much smaller than n.
1 Introduction
A coloring of a family S of n regions of some fixed type (such as discs, pseudodiscs, axisparallel
rectangles, etc.), is called conflictfree (CF for short) if for each point p bSb there is at least
one region b S that contains p in its interior, whose color is unique among all regions in S that
contain p in their interior (in this case we say that p is being `served' by that color).
The study of such colorings, which originated in [3] and [9], was motivated by the problem
of frequency assignment in cellular networks. Specifically, cellular networks are heterogeneous
networks with two different types of nodes: basestations (that act as servers) and clients. The
basestations are interconnected by an external fixed backbone network. Clients are connected
