Theorists Draw Closer to Perfect Coloring

A theorem for coloring a large class of “perfect” mathematical networks could ease the way for a long-sought general coloring proof.

Glen Faught for Quanta Magazine


Four years ago, the mathematician Maria Chudnovsky faced an all-too-common predicament: how to seat 120 wedding guests, some of whom did not get along, at a dozen or so conflict-free tables. Luckily, the problem fell squarely in her realm of expertise. She conceived of the guests as nodes in a network, with links between incompatible nodes. Her task was to color in the nodes using a spectrum of colors representing the different tables. As long as connected nodes never had the same color, there would be no drama at the reception.

Natalie Wolchover/Quanta Magazine

Maria Chudnovsky of Princeton University at the Association for Women in Mathematics Research Symposium in College Park, Md., in April.

As a master of this pursuit, known as “graph coloring,” Chudnovsky did the whole thing in her head and finished the seating chart in no time. “My husband was very impressed,” she said.

Networks of related objects, be they nodes or wedding guests, are known to mathematicians as “graphs,” and graph coloring is the much-studied act of partitioning these objects into conflict-free sets. Most graphs, with their tangle of interconnections, are impossible to color with a limited palette. The larger they are, the more colors you need. Moving from node to node, alternating between colors, you inevitably get into traffic jams that force you to pull new hues out of the box. Likewise, in the real world, seating charts, meeting schedules and delivery routes can seldom be made optimal. But since the 1960s, mathematicians have escaped these coloring frustrations by working with so-called perfect graphs, which “behave very nicely with respect to coloring,” said Chudnovsky, a 38-year-old math professor at Princeton University.

Perfect graphs are, by definition, colorable with the most limited palette possible. When coloring a graph, every node in a mutually connected cluster, or “clique,” must receive a distinct color, so any graph needs at least as many colors as the number of nodes in its largest clique. In most graphs, you need many more colors than this. But in perfect graphs, you do not. As the French graph theorist Claude Berge defined them in 1961, perfect graphs require a number of colors exactly equal to the size of their largest clique. The “chromatic number” must also equal the “clique number” for every subset of a perfect graph formed by deleting some of its nodes. This perfection rarely arises in the real world, but the property has made perfect graphs much easier to analyze and prove theorems about than their imperfect counterparts.

Yet, after half a century, an obvious question about perfect graphs remains unanswered: How do you actually color them? “Perfect graphs are the graphs that are designed to work well for coloring, so it’s really annoying that we don’t know a good way to color perfect graphs,” said Paul Seymour, a graph theorist also at Princeton. “For a mathematician, a problem like that is a magnet. You want to be able to fix the issue.”

Now, Chudnovsky and collaborators are taking significant steps toward a theorem for coloring all perfect graphs. They have spent the past few years “nibbling off different pieces of the pie,” said Alan Tucker, a mathematician at Stony Brook University, proving coloring theorems for ever-larger subclasses of perfect graphs. This month, in their most general result yet, Chudnovsky, together with Irene Lo,  Frédéric Maffray, Nicolas Trotignon and Kristina Vušković, posted a theorem for coloring all perfect graphs except those containing tricky arrangements of four nodes called “squares.” “It gives confidence that the general case might be solved,” said Gérard Cornuéjols, a mathematician at Carnegie Mellon University.

Andrew Silver for Quanta Magazine

Interactive: Select a color and then a node to color in this simple perfect graph. When the entire graph is colored in, “Check” that no connected nodes share the same color.

The hope is that history might repeat itself. Fifteen years ago, researchers raced to prove a theorem establishing the recipe for perfect graphs. After Cornuéjols, Vušković and Michele Conforti proved the theorem for “square-free” perfect graphs in 2001, “the general case came next,” Chudnovsky said.

It was in 2002 that Chudnovsky along with Seymour, then her Ph.D. advisor, and two more collaborators proved the “strong perfect graph theorem” establishing what it takes to be a perfect graph. Their proof, which was published in the Annals of Mathematics in 2006, filled 150 pages. But the strong perfect graph theorem provides a surprisingly simple recipe for perfection: As Berge correctly guessed 54 years ago, a graph is perfect when it does not contain any arrangements of five or more nodes called “odd holes” or “odd antiholes.”

Diagrams by Olena Shmahalo/Quanta Magazine

An odd hole is a closed-loop path through part of a graph that passes through an odd number of nodes. (If you drew the graph on paper and cut along this path with scissors, you would cut a hole in the paper.) In an odd antihole, the nodes are connected to all but their nearest neighbors, forming a star-like shape. To see why these oddities render graphs imperfect, consider, for instance, a “five-hole,” which looks like a pentagon: Its clique number is two, since only pairs of consecutive nodes are connected. But try to color the five-hole using only two colors — alternating, for instance, between blue and green — and you soon get into trouble: The fifth node has a blue neighbor on one side and a green neighbor on the other. A third color is needed. (Three-holes, unlike larger odd holes, are allowed to exist in perfect graphs, because their clique number is three.)

Real-world graphs such as conference schedules, the Manhattan subway system or the human neural network typically contain odd holes, making the study of perfect graphs primarily an intellectual exercise. And yet, “the class of perfect graphs allows you to develop sophisticated techniques that you can use in other classes,” said Vušković, a professor at the University of Leeds in the United Kingdom.

Even perfect graphs can be tremendously complex, demanding detailed consideration of each of their umpteen internal structures and seldom submitting to elegant, concise proofs. “The discrete pieces just don’t yield to overall theories,” Tucker said. In their new theorem for coloring all perfect graphs that lack squares (also known as “four-holes”), Chudnovsky, Lo, Maffray, Trotignon and Vušković took a “divide and conquer” approach, essentially breaking the graphs up into parts, coloring the parts, and then gluing them together again.

To color a given graph, their first step is to scour the graph for a structure called a “prism,” which consists of a pair of three-holes connected to each other via three paths.

Next, depending on how the prism attaches to the rest of the graph, the researchers partition the graph into two parts, left and right, with a set of nodes serving as a hinge between them. In general, this hinge might contain a square, but because there are too many possible ways to color hinges with squares, the current proof leaves out these tricky cases.

If either the left or right part contains another prism within it, the researchers must break it up again, and so on until no more prisms remain. (Here, graphs with squares again cause trouble, requiring too many partitions for the coloring procedure to work efficiently.)

Once neither left nor right contain a prism, then they can be colored in. The researchers proved that there is an efficient procedure for coloring both the left part and hinge together and the right part and hinge together. Typically, the two different colorings of the hinge won’t agree; a final step switches the colors of neighboring nodes until they match up.

Now, only cases with squares remain unsolved. Experts disagree about how close the researchers have come to a perfect graph coloring theorem. In Vušković’s opinion, “The square-free case of perfect graphs retains all the structural complexity of the perfect graph. It’s very close to the general case.” Cornuéjols, on the other hand, said, “I think it’s still a big step.”

The five collaborators will meet in Grenoble, France, in December to discuss ways to generalize their proof.

“We did a good step, but there are many steps more to be done,” said Trotignon, a mathematician and computer scientist at École Normale Superieure in Lyon, France. “My feeling now is that this problem will be solved. Before this step of square-free graphs, I would have said no.”

If the researchers succeed in proving a theorem for coloring all perfect graphs, some say it would mark the end of an era. “To me, that’s the last very big open question about them,” Cornuéjols said.

Correction: This article was revised on October 20, 2015, to reflect that the mathematician Claude Berge was from France, not Hungary.

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  • Once you have a way to color all perfect graphs, could you use it as a basis for coloring imperfect graphs? That is, could you remove the nodes that make a graph imperfect, color the resulting perfect graph, then add the nodes back and color them with the minimum number of extra colors? Or would some solutions of the perfect graph lead to sub-optimal solutions of the imperfect graph?

  • It's interesting how small the world of Mathematics is when viewed from the editorial office of Quanta Magazine.

  • I know articles need to start with a strong "lede" but impressing her husband with wedding planning skills? Is her life defined entirely by the men in it? Disappointed.

  • There's a delicious thrill in finally getting to use an unimaginably specialized skill ( in this case, graph theory) in one's private life. A claw-free, bull-free arrangement of wedding guests sounds like a worthy optimization problem, and would impress most spouses.

  • If i recall, the four color map conjecture "you can always color a map using at most four colors" was 'proven' in the 80's by exhaustively coloring all possible maps, using a computer. At the time, there was some grumbling about how this wasn't a proof. My guess is that doing it this way wasn't very instructive. This sort of analysis does appear to Map makers had known that you didn't need more than four colors for centuries.

    There are issues with a map of the world that violate the premise. In particular, there are countries that aren't contiguous. The USA has Alaska and Hawaii. There are other exceptions.

  • Don't computer algorithms for quantum electrodynamics and quantum chromodynamics have to solve such problems on their way to solving their bigger problems, such as the expected ratios of particle productions at particle colliders? Hasn't a great deal of work been done by physicists on this area already? Do the computer simulation designers at CERN, Fermilab, SLAC, etc and the graph theory people collaborate?

  • @Fred With computer simulation you can only do finite cases. Computers can be useful in math when you found a way to prove that for very large integers (or number of nodes here) you have the property you desire. Then the computers are just there to fill the (finite) gap. I think this is what happened for the four color problem. In the case of the proof of the ternary Goldbach conjecture by Harald Helfgott, there already was a result for very large integers but the gap was still too huge to fill with computers. He brought down the required computations by order of magnitudes and the gap was then filled using computers. The mathematical community now accepts computer help more than at the time of the four color conjecture.

  • I can not see any difference between an "odd hole" and an "odd antihole". It is the same graph, drawn two ways.

  • Could you send me the next page ? I feel I got lucky with this one.. Even number of holes in a circle can always be solved with two colors as long as they are all connected at both ends. The problem at top is an even number of holes but all are not connected at both ends.

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