Ronald Graham

Ronald Lewis Graham (October 31, 1935 – July 6, 2020)[1] was an American mathematician credited by the American Mathematical Society as "one of the principal architects of the rapid development worldwide of discrete mathematics in recent years".[2]

Ronald Graham
Ronald graham writing.jpg
Graham in 1998
Ronald Lewis Graham

(1935-10-31)October 31, 1935
DiedJuly 6, 2020(2020-07-06) (aged 84)
San Diego, California, U.S.
Alma mater
Known for
Spouse(s)Fan Chung
Scientific career
ThesisOn Finite Sums of Rational Numbers (1962)
Doctoral advisorDerrick Henry Lehmer

He did important work in scheduling theory, computational geometry, Ramsey theory, and quasi-randomness.[3] He worked for many years at Bell Labs and later at the University of California, San Diego, and was president of both the American Mathematical Society and the Mathematical Association of America.

Ronald Graham juggling a four-ball fountain (1986)

Graham has been featured in Ripley's Believe It or Not! for being not only "one of the world's foremost mathematicians", but also an accomplished trampolinist and juggler, and in 1972 was elected president of the International Jugglers' Association.[4][5][3]


Graham was born in Taft, California, on October 31, 1935,[6] the son of an oil field worker and later merchant marine. Despite his later interest in gymnastics, he was small and non-athletic.[7] He grew up moving frequently between California and Georgia, skipping several grades of school in these moves, and never staying at any one school longer than a year.[1][7] As a teenager, he moved to Florida with his now-divorced mother, where he went to but did not finish high school. Instead, at the age of 15 he won a Ford Foundation scholarship to the University of Chicago, where he learned gymnastics but no mathematics.[1]

After three years, when his scholarship expired, he moved to the University of California, Berkeley, officially as a student of electrical engineering but also studying number theory under Derrick Henry Lehmer,[1] and winning a title as California state trampoline champion.[7] He enlisted in the United States Air Force in 1955, when he reached the age of eligibility,[8] left Berkeley without a degree, and was stationed in Fairbanks, Alaska, where he finally completed a bachelor's degree in physics in 1959 at the University of Alaska Fairbanks.[1] Returning to the University of California, Berkeley for graduate study, he received his Ph.D. in mathematics in 1962. His dissertation, supervised by Lehmer, was On Finite Sums of Rational Numbers.[9] While a graduate student, he supported himself by performing on trampoline in a circus,[8] and married Nancy Young, an undergraduate mathematics student at Berkeley; they had two children.[1]

Ronald Graham, his wife Fan Chung, and Paul Erdős, Japan 1986

After completing his doctorate, Graham began working in 1962 at Bell Labs and (as it later became) AT&T Labs, in New Jersey, as Director of Information Sciences. In 1963, at a conference in Colorado, he met the prolific Hungarian mathematician Paul Erdős (1913−1996),[1] who became a close friend and frequent research collaborator. Graham was chagrined to be beaten in ping-pong by Erdős, then already middle-aged; he returned to New Jersey determined to improve his game, and eventually became Bell Labs champion and won a state title in the game.[1] Graham later popularized the concept of the Erdős number, a measure of distance from Erdős in the collaboration network of mathematicians;[10][8] his many works with Erdős include two books of open problems[B1][B5] and Erdős's final posthumous paper.[A15] Graham divorced in the 1970s; in 1983 he married his Bell Labs colleague and frequent coauthor Fan Chung.[1]

While at Bell Labs, Graham also took a position at Rutgers University as University Professor of Mathematical Sciences in 1986, and served a term as president of the American Mathematical Society from 1993 to 1994. He became Chief Scientist of the labs in 1995.[1] He retired from AT&T in 1999 after 37 years of service there,[11] and moved to the University of California, San Diego (UCSD), as the Irwin and Joan Jacobs Endowed Professor of Computer and Information Science.[1][8] At UCSD, he also became chief scientist at the California Institute for Telecommunications and Information Technology.[8][5] In 2003−04, he was president of the Mathematical Association of America.[1]

Graham died of bronchiectasis[12] on July 6, 2020, aged 84, in La Jolla, California.[6][13]


Graham made important contributions in multiple areas of mathematics and theoretical computer science. He published over 350 papers[13] and six books, including Concrete Mathematics with Donald Knuth and Oren Patashnik,[B4] and the Erdős Number Project lists him as having nearly 200 coauthors.[14]

Notable topics in mathematics named after Graham include the Erdős–Graham problem on Egyptian fractions, the Graham–Rothschild theorem in the Ramsey theory of parameter words and Graham's number derived from it, the Graham–Pollak theorem and Graham's pebbling conjecture in graph theory, the Coffman–Graham algorithm for approximate scheduling and graph drawing, and the Graham scan algorithm for convex hulls. He also began the study of primefree sequences, the Boolean Pythagorean triples problem, the biggest little polygon, and square packing in a square.

As well as publishing under his own name, Graham has participated in the publications of G. W. Peck, a pseudonymous mathematical collaboration named for the initials of its members, with Graham as the "G".[15]

Number theoryEdit

Graham's doctoral dissertation was in number theory, on Egyptian fractions,[7][9] and the Erdős–Graham problem is closely related. It asked for a proof that, when the integers are partitioned into finitely many classes, one of the classes has a subset whose reciprocals sum to one. A proof was published by Ernie Croot in 2003.[16] Another of Graham's papers on Egyptian fractions was published in 2015 with Steve Butler and (nearly 20 years posthumously) Paul Erdős; it was the last of Erdős's papers to be published, making Butler his 512th coauthor.[A15][17]

In a 1964 paper, Graham began the study of primefree sequences by observing that there exist sequences of numbers, defined by the same recurrence relation as the Fibonacci numbers, in which none of the sequence elements is prime.[A64] The challenge of constructing more such sequences was later taken up by Donald Knuth and others.[18] Graham's 1980 book with Paul Erdős, Old and new results in combinatorial number theory, provides a collection of open problems from a broad range of subareas within number theory.[B1]

Ramsey theoryEdit

The Graham–Rothschild theorem in Ramsey theory was published by Graham and Bruce Rothschild in 1971, and applies Ramsey theory to combinatorial cubes in combinatorics on words.[A71a] Graham gave a large number as an upper bound for an instance of this theorem, now known as Graham's number, which was listed in the Guinness Book of Records as the largest number ever used in a mathematical proof,[19] although it has since then been surpassed by even larger numbers such as TREE(3).[20]

Graham offered a monetary prize for solving the Boolean Pythagorean triples problem, another problem in Ramsey theory; the prize was claimed in 2016.[21] Graham also published two books on Ramsey theory.[B2][B3]

Graph theoryEdit

Partition of the edges of the complete graph   into five complete bipartite subgraphs, according to the Graham–Pollak theorem

The Graham–Pollak theorem, which Graham published with Henry O. Pollak in two papers in 1971 and 1972,[A71b][A72a] states that if the edges of an  -vertex complete graph are partitioned into complete bipartite subgraphs, then at least   subgraphs are needed. Graham and Pollak provided a simple proof using linear algebra, and despite the combinatorial nature of the statement and despite multiple publications of alternative proofs since their work, all known proofs require linear algebra.[22]

Soon after research in quasi-random graphs began with the work of Andrew Thomason, Graham and his coauthors Fan Chung and R. M. Wilson published in 1989 a result that has been called the "fundamental theorem of quasi-random graphs", stating that many different definitions of these graphs are equivalent.[A89a][23]

Graham's pebbling conjecture, appearing in a 1989 paper by Fan Chung, concerns the pebbling number of Cartesian products of graphs. As of 2019, it remains unsolved.[24]

Packing, scheduling, and approximation algorithmsEdit

Graham's early work on job shop scheduling[A66][A69] introduced the worst-case approximation ratio into the study of approximation algorithms, and laid the foundations for the later development of competitive analysis of online algorithms.[25] This work was later recognized to be important also for the theory of bin packing,[26] an area that Graham later worked in more explicitly.[A74]

The Coffman–Graham algorithm, which Graham published with Edward G. Coffman Jr. in 1972,[A72b] provides an optimal algorithm for two-machine scheduling, and a guaranteed approximation algorithm for larger numbers of machines. It has also been applied in layered graph drawing.[27]

In a survey article on scheduling articles published in 1979, Graham and his coauthors introduced a three-symbol notation for classifying theoretical scheduling problems according to the system of machines they are to run on, the characteristics of the tasks and resources such as requirements for synchronization or non-interruption, and the performance measure to be optimized.[A79] This classification has sometimes been called "Graham notation" or "Graham's notation".[28]

Discrete and computational geometryEdit

The Graham scan algorithm for convex hulls

Graham scan is a widely used and practical algorithm for convex hulls of two-dimensional point sets, based on sorting the points and then inserting them into the hull in sorted order.[29] Graham published the algorithm in 1972.[A72c]

The biggest little polygon problem asks for the polygon of largest area for a given diameter. Surprisingly, as Graham observed, the answer is not always a regular polygon.[A75a] Graham's 1975 conjecture on the shape of these polygons was finally proven in 2007.[30]

In another 1975 publication, Graham and Erdős observed that for packing unit squares into a larger square with non-integer side lengths, one can use tilted squares to leave an uncovered area that is sublinear in the side length of the larger square, unlike the obvious packing with axis-aligned squares.[A75b] Klaus Roth and Bob Vaughan proved that uncovered area at least proportional to the square root of the side length may sometimes be needed; proving a tight bound on the uncovered area remains an open problem.[31]

Probability and statisticsEdit

In nonparametric statistics, a 1977 paper by Persi Diaconis and Graham studied the statistical properties of Pearson's footrule, a measure of rank correlation that compares two permutations by summing, over each item, the distance between the positions of the item in the two permutations.[A77] They compared this measure to other rank correlation methods, resulting in the "Diaconis–Graham inequalities"


where   is Pearson's footrule,   is the number of inversions between the two permutations (a non-normalized version of the Kendall rank correlation coefficient), and   is the minimum number of two-element swaps needed to obtain one permutation from the other.[32]

The Chung–Diaconis–Graham random process is a random walk on the integers modulo an odd integer  , in which at each step one doubles the previous number and then randomly adds zero,  , or   (modulo  ). In a 1987 paper, Fan Chung, Diaconis, and Graham studied the mixing time of this process, motivated by the study of pseudorandom number generators.[A87][33]

Awards and honorsEdit

In 2003, Graham won the American Mathematical Society's annual Leroy P. Steele Prize for Lifetime Achievement. The prize cited his contributions to discrete mathematics, his popularization of mathematics through his talks and writing, his leadership at Bell Labs, and his service as president of the society.[34] He was one of five inaugural winners of the George Pólya Prize of the Society for Industrial and Applied Mathematics, sharing it with fellow Ramsey theorists Klaus Leeb, Bruce Rothschild, Alfred Hales, and Robert I. Jewett.[35] He was also one of two inaugural winners of the Euler Medal of the Institute of Combinatorics and its Applications, the other being Claude Berge.[36]

Graham was elected to the National Academy of Sciences in 1985.[37] In 1999 he was inducted as an ACM Fellow "for seminal contributions to the analysis of algorithms, in particular the worst-case analysis of heuristics, the theory of scheduling, and computational geometry".[38] He became a Fellow of the Society for Industrial and Applied Mathematics in 2009; the fellow award cited his "contributions to discrete mathematics and its applications".[39] In 2012 he became a fellow of the American Mathematical Society.[40]

Graham was an invited speaker at the 1982 International Congress of Mathematicians (held 1983 in Warsaw),[13] speaking on "Recent developments in Ramsey theory".[A84] He was twice Josiah Willard Gibbs Lecturer, in 2001 and 2015.[13] The Mathematical Association of America awarded him both the Carl Allendoerfer Prize for his paper "Steiner Trees on a Checkerboard" with Fan Chung and Martin Gardner in Mathematics Magazine (1989),[A89b][41] and the Lester R. Ford Award for his paper "A whirlwind tour of computational geometry" with Frances Yao in the American Mathematical Monthly (1990).[A90][42] His book Magical Mathematics with Persi Diaconis[B6] won the Euler Book Prize.[43]

The proceedings of the Integers 2005 conference was published as a festschrift for Ron Graham's 70th birthday.[44] Another festschrift, stemming from a conference held in 2015 in honor of Graham's 80th birthday, was published in 2018 as the book Connections in discrete mathematics: a celebration of the work of Ron Graham.[45]

Selected publicationsEdit


B1. Old and new results in combinatorial number theory. With Paul Erdős. Monographie 28, L'Enseignement Mathématique, 1980.[46]
B2. Ramsey Theory. With Bruce Rothschild and Joel Spencer. Wiley, 1980; 2nd ed., 1990.[47]
B3. Rudiments of Ramsey Theory. American Mathematical Society, 1981; 2nd ed., with Steve Butler, 2015.[48]
B4. Concrete Mathematics: a foundation for computer science. With Donald Knuth and Oren Patashnik. Addison-Wesley, 1989; 2nd ed., 1994.[49]
B5. Erdős on Graphs. His legacy of unsolved problems. With Fan Chung. A K Peters, 1998.[50]
B6. Magical Mathematics: the mathematical ideas that animate great magic tricks. With Persi Diaconis. Princeton University Press, 2011.[51]

Edited volumesEdit

V1. Handbook of Combinatorics. Edited with Martin Grötschel and László Lovász. MIT Press, 1995.[52]
V2. The mathematics of Paul Erdős. Edited with Jaroslav Nešetřil. 2 volumes. Springer, 1997; 2nd ed., 2013.[53]


A64. Graham, Ronald L. (1964). "A Fibonacci-like sequence of composite numbers" (PDF). Mathematics Magazine. 37 (5): 322–324. doi:10.2307/2689243. JSTOR 2689243. MR 1571455.
A66. Graham, R. L. (1966). "Bounds for certain multiprocessing anomalies" (PDF). Bell System Technical Journal. 45 (9): 1563–1581. doi:10.1002/j.1538-7305.1966.tb01709.x.
A69. Graham, R. L. (1969). "Bounds on multiprocessing timing anomalies" (PDF). SIAM Journal on Applied Mathematics. 17: 416–429. doi:10.1137/0117039. MR 0249214.
A71a. Graham, R. L.; Rothschild, B. L. (1971). "Ramsey's theorem for n-parameter sets" (PDF). Transactions of the American Mathematical Society. 159: 257–292. doi:10.1090/S0002-9947-1971-0284352-8. JSTOR 1996010. MR 0284352.
A71b. Graham, R. L.; Pollak, H. O. (1971), "On the addressing problem for loop switching" (PDF), Bell System Technical Journal, 50: 2495–2519, doi:10.1002/j.1538-7305.1971.tb02618.x, MR 0289210
A72a. Graham, R. L.; Pollak, H. O. (1972), "On embedding graphs in squashed cubes", Graph theory and applications (Proc. Conf., Western Michigan Univ., Kalamazoo, Mich., 1972; dedicated to the memory of J. W. T. Youngs) (PDF), Lecture Notes in Mathematics, 303, pp. 99–110, MR 0332576
A72b. Coffman, E. G. Jr.; Graham, R. L. (1972). "Optimal scheduling for two-processor systems" (PDF). Acta Informatica. 1: 200–213. doi:10.1007/bf00288685. MR 0334913.
A72c. Graham, R. L. (1972). "An efficient algorithm for determining the convex hull of a finite planar set" (PDF). Information Processing Letters. 1 (4): 132–133. doi:10.1016/0020-0190(72)90045-2.
A74. Johnson, D. S.; Demers, A.; Ullman, J. D.; Garey, M. R.; Graham, R. L. (1974). "Worst-case performance bounds for simple one-dimensional packing algorithms" (PDF). SIAM Journal on Computing. 3: 299–325. doi:10.1137/0203025. MR 0434396.
A75a. Graham, R. L. (1975). "The largest small hexagon" (PDF). Journal of Combinatorial Theory. Series A. 18: 165–170. doi:10.1016/0097-3165(75)90004-7. MR 0360353.
A75b. Erdős, P.; Graham, R. L. (1975). "On packing squares with equal squares" (PDF). Journal of Combinatorial Theory. Series A. 19: 119–123. doi:10.1016/0097-3165(75)90099-0. MR 0370368.
A77. Diaconis, Persi; Graham, R. L. (1977). "Spearman's footrule as a measure of disarray". Journal of the Royal Statistical Society. 39 (2): 262–268. doi:10.1111/j.2517-6161.1977.tb01624.x. JSTOR 2984804. MR 0652736.
A79. Graham, R. L.; Lawler, E. L.; Lenstra, J. K.; Rinnooy Kan, A. H. G. (1979). "Optimization and approximation in deterministic sequencing and scheduling: a survey" (PDF). Annals of Discrete Mathematics. 5: 287–326. MR 0558574.
A84. Graham, R. L. (1984). "Recent developments in Ramsey theory" (PDF). Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983). Warsaw: PWN. pp. 1555–1567. MR 0804796.
A87. Chung, F. R. K.; Diaconis, Persi; Graham, R. L. (1987), "Random walks arising in random number generation" (PDF), Annals of Probability, 15 (3): 1148–1165, JSTOR 2244046, MR 0893921
A89a. Chung, F. R. K.; Graham, R. L.; Wilson, R. M. (1989). "Quasi-random graphs" (PDF). Combinatorica. 9 (4): 345–362. doi:10.1007/BF02125347. MR 1054011.
A89b. Chung, Fan; Gardner, Martin; Graham, Ron (1989). "Steiner trees on a checkerboard" (PDF). Mathematics Magazine. 62 (2): 83–96. doi:10.2307/2690388. JSTOR 2690388. MR 0991536.
A90. Graham, Ron; Yao, Frances (1990). "A whirlwind tour of computational geometry" (PDF). American Mathematical Monthly. 97 (8): 687–701. doi:10.2307/2324575. JSTOR 2324575. MR 1072812.
A15. Butler, Steve; Erdős, Paul; Graham, Ron (2015). "Egyptian fractions with each denominator having three distinct prime divisors" (PDF). Integers. 15: A51. MR 3437526.


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  22. ^ Aigner, Martin; Ziegler, Günter M. (2018). Proofs from THE BOOK (6th ed.). Springer. pp. 79–80. doi:10.1007/978-3-662-57265-8_15. ISBN 978-3-662-57265-8.
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  24. ^ Pleanmani, Nopparat (2019). "Graham's pebbling conjecture holds for the product of a graph and a sufficiently large complete bipartite graph". Discrete Mathematics, Algorithms and Applications. 11 (6): 1950068, 7. doi:10.1142/s179383091950068x. MR 4044549.
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  27. ^ Bastert, Oliver; Matuszewski, Christian (2001). "Layered drawings of digraphs". In Kaufmann, Michael; Wagner, Dorothea (eds.). Drawing Graphs: Methods and Models. Lecture Notes in Computer Science. 2025. Springer-Verlag. pp. 87–120. doi:10.1007/3-540-44969-8_5.
  28. ^ For a recent example, see e.g. Cygan, Marek; Pilipczuk, Marcin; Pilipczuk, Michał; Wojtaszczyk, Jakub Onufry (2014). "Scheduling partially ordered jobs faster than  ". Algorithmica. 68 (3): 692–714. doi:10.1007/s00453-012-9694-7. MR 3160651.
  29. ^ De Berg, Mark; Cheong, Otfried; Van Kreveld, Marc; Overmars, Mark (2008). Computational Geometry Algorithms and Applications. Berlin: Springer. pp. 2–14. doi:10.1007/978-3-540-77974-2. ISBN 978-3-540-77973-5.
  30. ^ Foster, Jim; Szabo, Tamas (2007). "Diameter graphs of polygons and the proof of a conjecture of Graham". Journal of Combinatorial Theory. Series A. 114 (8): 1515–1525. doi:10.1016/j.jcta.2007.02.006. MR 2360684..
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  33. ^ Hildebrand, Martin (2019). "On a lower bound for the Chung-Diaconis-Graham random process". Statistics & Probability Letters. 152: 121–125. doi:10.1016/j.spl.2019.04.020. MR 3953053.
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  38. ^ "Ronald L. Graham". ACM Fellows. Association for Computing Machinery. Retrieved July 12, 2020.
  39. ^ "SIAM Fellows". Society for Industrial and Applied Mathematics. Retrieved July 11, 2020.
  40. ^ "List of Fellows of the American Mathematical Society". American Mathematical Society. Retrieved July 9, 2020.
  41. ^ "Allendoerfer Award". MAA Awards. Mathematical Association of America. Retrieved July 9, 2020.
  42. ^ "Paul R. Halmos - Lester R. Ford Awards". MAA Awards. Mathematical Association of America. Retrieved July 9, 2020.
  43. ^ "Euler Book Prize" (PDF). MAA Prizes Awarded in San Diego. Notices of the American Mathematical Society. 60 (5): 613–614. May 2013.
  44. ^ Proceedings of the Integers Conference 2005 in honor of Ron Graham's 70th birthday. Carrollton, GA: Integers. 2007. MR 2395797.
  45. ^ Butler, Steve; Cooper, Joshua; Hurlbert, Glenn, eds. (2018). Connections in discrete mathematics: a celebration of the work of Ron Graham. Cambridge University Press. ISBN 978-1-316-60788-6. Reviewed by Hopkins, David (June 2019). The Mathematical Gazette. 103 (557): 374–375. doi:10.1017/mag.2019.82.CS1 maint: untitled periodical (link)
  46. ^ Review of Old and new problems and results in combinatorial number theory:
  47. ^ Reviews of Ramsey Theory:
  48. ^ Reviews of Rudiments of Ramsey Theory:
  49. ^ Reviews of Concrete Mathematics:
  50. ^ Reviews of Erdős on Graphs:
  51. ^ Reviews of Magical Mathematics:
  52. ^ Reviews of Handbook of Combinatorics:
  53. ^ Reviews of The Mathematics of Paul Erdős:

External linksEdit