Ronald L. Graham, who unlocked the magic of numbers, dies at 84

Ronald L. Graham, who gained renown with wide-ranging theorems in a field known as discrete mathematics that have found uses in diverse areas, from making phone and computer networks more efficient to explaining the dynamics of juggling, died July 6 at his home in San Diego. He was 84.

The cause was bronchiectasis, a chronic lung condition, according to a statement from the University of California San Diego, where Mr. Graham was an emeritus professor.

“He created a lot of mathematics and some really pretty cool stuff,” said Peter Winkler, a mathematician at Dartmouth College. “This occurred over many years, and so it’s only now that we get to sort of look back and see all the stuff that he did.”


One thing he did was develop methods for worst-case analysis in scheduling theory — that is, whether the order in which actions are scheduled wastes time. On another front, with his wife and frequent collaborator, Fan Chung, an emeritus mathematician at the University of California San Diego, he developed the idea of quasi-random graphs, which applied numerical preciseness in describing the random-like structure of networks.

Mr. Graham’s research was detailed in about 400 papers, but he never fit the stereotype of a nerdy mathematician. Soft-spoken but garrulous, he leavened his talks on high-level equations with jokes and sight gags. He was also an expert trampoline gymnast and juggler, a pursuit that in his hands also lent itself to mathematical analysis. At one point Mr. Graham and three other juggling mathematicians proved an equation for the number of possible ball-juggling patterns before a pattern repeats.

Mr. Graham was a collaborator and close friend of Paul Erdös, one of the great mathematicians of the 20th century. Erdös cared only about numbers, so much so that he lived without a permanent home or job. Carrying a single piece of battered luggage, he would flit from one place to another, relying on the hospitality of colleagues, including Mr. Graham.


Erdös was not, however, the easiest of houseguests. “After a couple of days, they start fighting,” Chung said of Erdös and her husband. When they met, Mr. Graham and Erdös were among the few working in discrete mathematics, particularly in combinatorics — the math of combinations.

In an introductory probability class, a simple combinatorics problem might be: If one pulls three balls at random out of a bag of six blue ones and four red ones, what are the chances that all three are red? (The answer is 1 out of 30.)

Combinatorics proved to be important to the rise of digital technology in the 1970s.

It led to what became known as Graham’s number, which was for a time the largest number used in a proof, according to the Guinness Book of World Records. The number came out of a problem known as the Ramsey theory, which states that in large systems there can never be complete disorder, that pockets of structure will appear.

Ronald Lewis Graham was born to Leo and Margaret Jane (Anderson) Graham on Oct. 31, 1935, in Taft, Calif. His father worked in the oil fields, and both parents later worked in shipyards, moving with the family back and forth between California and Georgia, resulting in Ronald’s skipping several grades. After his parents divorced, he and his mother moved to Florida.


Without graduating from high school, Mr. Graham received a Ford Foundation scholarship to the University of Chicago at 15. When his scholarship ran out, he transferred to the University of California Berkeley, where he majored in electrical engineering and studied number theory.

In 1955, he enlisted in the Air Force and was assigned to Alaska. He signed up to work the night shift so that he could attend the University of Alaska. He received his bachelor’s degree in 1958 in physics, because the university was not accredited to award math degrees.

He returned to Berkeley for graduate school, where he and two friends formed a professional trampoline group, which performed with a circus.

After obtaining a doctoral degree in mathematics from Berkeley in 1962, Mr. Graham joined Bell Labs, solving problems that proved helpful for a telephone company. In the 1960s, a Bell Labs engineer named John R. Pierce came up with the idea of dividing up how phone calls were sent from one place to another, a precursor to what is now known as packet switching.

“Until then, communication was done by phone lines, and lines had to be open from one end to the other,” Winkler said.

In Pierce’s method, the data carrying the sound of a phone call was chopped apart, and “that information would be piled into these little packets, and these packets would swim around the network” — more efficient, since one line could now handle many calls at once.

In addition to Chung, whom he married in 1983, survivors include a son, Marc; three daughters, Ché Graham, Christy Newman and Laura Lindauer; two stepchildren, Dean Chung and Laura Bower; a brother, Jerry Graham; and 11 grandchildren.