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Graph Theory Uncovers the Roots of Perfection

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Graph Theory Uncovers the Roots of Perfection N EWS F OCUS 65 were later named “Berge graphs”) would be 64 MATHEMATICS perfect (see figure). But he couldn’t prove it, 63 and his speculation became the SPGC. At 62 the same time, Berge also ventured a less 61 Graph Theory Uncovers the definitive pronouncement known as the 60 “weak” perfect graph conjecture. 59 Seymour, along with co-authors G. Neil 58 Roots of Perfection Robertson of Ohio State University in 57 Columbus and Robin Thomas of the Georgia A newly minted proof tells how to recognize which arrangements of 56 Institute of Technology in Atlanta, began points and lines are the crème de la crème 55 working on the SPGC in 2000. They were 54 motivated in part by a grant from the private- 53 To some, perfection is priceless. But for phone network based on a perfect graph ly funded American Institute of Mathematics 52 four graph theorists, it has a very specific could run with optimum efficiency even if in Palo Alto, California, which promotes 51 value. If their solution to one of the oldest some of its transmitters were knocked out work on high-profile unsolved problems. Al- 50 problems in their discipline—a classifica- (although it would cover less area). though mathematicians have identified a 49 tion of so-called perfect graphs—holds up, A perfect graph is like a perfect chocolate dizzying variety of perfect graphs—96 types 48 they will reap a $10,000 bounty. cake: It might be easy to describe, but it’s hard at last count—the three researchers focused 47 The strong perfect graph conjecture to produce a recipe. In 1960, however, Claude on just two types, called bipartite graphs and 46 (SPGC) has perplexed mathematicians for Berge, a mathematician at the Centre National line graphs, as well as their antigraphs. Fol- 45 more than 40 years. “It’s a problem that ev- de la Recherche Scientifique in Paris, did just lowing a strategy suggested by another per- 44 eryone in graph theory knows about, and that. He noticed that every imperfect graph he fect-graph aficionado, Gerard Cornuejols of 43 some people in related areas, particularly could find contained either an “odd hole” or Carnegie Mellon University in Pittsburgh, 42 linear programming,” says Paul Seymour of an “odd anti-hole.” An odd hole is a ring of an Pennsylvania, they proved that any Berge 41 Princeton University, who announced the odd number (at least 5) of nodes, each linked graph that is not one of these types can be 40 proof at a meeting of the Canadian Mathe- to its two neighbors but not to any other node decomposed into smaller pieces that are. 39 matical Society last month. Its solution in the ring. An anti-hole is the reverse: Each Cornuejols had done more than just sug- 38 might enable mathematicians to quickly node is connected to every other node in the gest a strategy. He put his money where his 37 identify perfect graphs, which have proper- ring except its neighbors. mouth is, offering $5000 for the proof of a 36 ties that make otherwise intractable prob- Berge boldly conjectured that any graph particular step that had eluded him, as well 35 lems involving networks easy to solve. that avoided these two flaws (such graphs as $5000 for completing the proof of the full 34 The graphs in question consist of noth- SPGC. To collect the prize, Seymour and his 33 ing more than dots and lines. Each line con- collaborators will have to get their work pub- 32 nects exactly two dots, or nodes. The SPGC lished, which might take a year or more 31 grew out of mathematicians’ fascination channels clique while other graph theorists scrutinize a proof 30 with coloring graphs in such a way that no that will likely run to 150 or 200 pages. 29 two nodes of the same color are connected, The early betting is that they will collect 28 a problem rooted in the real-world business the prize. “I don’t know the details of the 27 of coloring maps. When Wolfgang Haken proof, but I trust that it is basically correct,” 26 and Kenneth Appel proved the famous Four- says László Lovász of Microsoft Research, 25 Color Theorem for planar maps in 1976, who proved the weak perfect graph theorem 24 they did it by means of graph theory. chi = omega = 3 in 1972. “In the first version of a complicat- 23 Coloring problems make sense for other ed and long proof like this, there are always 22 kinds of graphs as well. In a cell-phone some gaps and, at the same time, often sub- 21 network, for example, the nodes are trans- perfect stantial possibilities for simplification. But I 20 mitters, the lines connect any two transmit- do know the general plan, and I have no 19 ters whose ranges overlap, and the colors 12 doubt that it is now working.” Late last 18 correspond to channels. Coloring the net- month, Seymour and his student Maria 13 3 17 work amounts to assigning channels so that 12 Chudnovsky presented more details at a 16 no adjacent transmitters broadcast on the workshop in Oberwolfach, Germany. András 11 4 15 same channel. Of course, the phone compa- chi = 4 Sebö of the Institut d’Informatique et Mathé- 14 ny would want to use the smallest possible omega = 3 matiques Appliquées in Grenoble, France, 13 number of channels, which is called the 10 5 who co-organized the workshop, says the ex- 12 (χ) chromatic number chi of the network. 9 6 cellent track record of the authors, and the 11 It’s easy to see that any group of nodes 8 7 ease with which Seymour and Chudnovsky 10 that are all connected to one another must all fielded all questions, leaves “not much 9 be different colors. Graph theorists call such imperfect doubt” that the proof will hold up. 8 a dense web of nodes a clique. Thus, in any Before the workshop, Seymour e-mailed 7 graph, chi has to be at least as large as the Net watch. In a web of cell-phone transmit- news of the proof to Berge, who proposed the 6 size of the biggest clique, a number known ters based on a perfect graph, the smallest problem. He later heard that Berge, who is ω 5 as omega ( ). In a perfect graph, in fact, chi number of channels needed to avoid interfer- seriously ill, had the message read to him in 4 and omega are exactly equal. More than that, ence (chi) equals the largest number of inter- the hospital. “He was happy,” Seymour says. 3 they stay equal no matter how many nodes connected nodes (omega). Adding two trans- –DANA MACKENZIE 2 you knock out of the graph, as long as the re- mitters that create an “odd hole” (arrows) Dana Mackenzie is a writer in Santa Cruz, 1 maining nodes keep their links intact. A cell- makes the graph imperfect. California. CREDIT: GRAPH BY COLIN MCDIARMID/UNIVERSITY OF OXFORD FROM ADAPTED 38 5 JULY 2002 VOL 297 SCIENCE www.sciencemag.org.
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