We prove that any collection of n discs in which each one intersects at most k others, can be colored with at most O(log 3 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.
Alon and Smorodinsky REF extend this by showing that each family of disks, where each disk intersects at most k others, can be colored using O(log 3 k) colors.
Conflict-free colorings of shallow discs
SCG '06
Mathematics Computer Science