Mathematicians find new answers to the still puzzling theorem that four colors suffice to color any map.
In 1852, botanist Francis Guthrie noticed something peculiar as he was coloring a map of counties in England. Despite the counties’ meandering shapes and varied configurations, four colors were all he needed to shade the map so that any two bordering counties were different colors. Perhaps, he speculated, four colors were enough for any map.
Little did Guthrie know the load of trouble he unleashed with his innocent conjecture. It took mathematicians nearly a century and a quarter to prove him right, and even that wasn’t enough to close the Pandora’s box Guthrie had opened. Mathematicians pulled out their markers and tried to color everything in sight.
The particular things mathematicians wanted to color were graphs: dots connected by lines Such graphs can be used to describe everything from friendships to the Internet to gene interactions. They can even describe maps, if the countries correspond to dots and bordering countries are connected by lines. Graphs from maps have the special property that the lines will never cross, though other graphs can form hairballs as nasty as you please. How many colors, mathematicians wondered, would it take to color any graph so that connected dots are always different colors?
That question turns out to be surprisingly important, and not just to a few marker-crazed mathematicians. Cell phone companies, for example, need to assign separate channels to any two transmitters whose ranges overlap in order to avoid interference. Naturally, they’d like to use the smallest number of channels for the job. Turn the transmitters into dots (called nodes), connect the nodes with a line (called an edge) if the ranges overlap, imagine the channels as colors assigned to the nodes, and voila! The phone companies are trying to solve a graph coloring problem.
Unfortunately, the graph coloring problem is nearly impossible to answer in full generality. But after decades of effort, a team of mathematicians has managed to characterize one major class of graphs for which they can solve the problem :the graphs that are “perfect.”
The mathematicians started their analysis with a simple observation: suppose there is a group of nodes in which every node is connected to every other one. Then every node in that group will need to be a different color. That means that a graph will need at least as many colors as there are members in the largest such interconnected group, and sometimes more. If this number of colors (called the clique number) is enough, the graph is called perfect. (Technically, there’s one more requirement for perfection: If you knock any number of nodes out of the graph along with all the edges that connect to those nodes, the clique number must remain sufficient to color the new graph.)
Easy to say, but which graphs are perfect?
This puzzle is one graph theorists have worried on for decades. They started their puzzling by looking for the “flaws” that make some graphs imperfect. One imperfect graph is a ring with an odd number of nodes, at least five, with each node connected just to its nearest neighbors. In this “odd hole,” only two nodes form a clique, but three colors are needed to color it. Another way to have an imperfect graph is the reverse, an “odd anti-hole”: Take a ring with an odd number of nodes, at least five, and connect each node to every other except for its neighbors.
The late mathematician Claude Berge of CNRS in Paris noticed that every imperfect graph he could find contained one of these two flaws. Those must be the only ones, he guessed, and in 1960 he created the strong perfect graph conjecture, the claim that a graph with neither an odd hole nor an odd anti-hole is perfect.
But he couldn’t prove it. And for nearly 40 years, the matter rested there.
In 2006, the problem finally yielded its secrets, and Paul Seymour of Princeton University, Robin Thomas of the Georgia Institute of Technology, Neil Robertson of Ohio State University, and Maria Chudnovsky of Columbia University published a proof. Chudnovsky presented the work at the Joint Mathematics Meetings in Washington, D.C. in January.
“It’s a brilliant proof,” says Gerard Cornuejols of Carnegie Mellon University and Universite d'Aix-Marseille, whose ideas the team built upon. To encourage work on the problem, he offered $5,000 of his own money for a proof, which the team collected.
Even with the proof, Guthrie’s Pandora’s box hasn’t been shut. Many more kinds of graphs remain to be colored. Furthermore, now mathematicians are looking for algorithms to efficiently detect perfect graphs. Others are seeking methods to find the minimal coloring the strong perfect graph theorem has shown must exist. The problems go on and on and on …
Chudnovsky, M., et al. 2006. The strong perfect graph theorem. Annals of Math 164:51-229. [Go to].