# Claude Jacques Roger Berge

### Born: 5 June 1926 in Paris, France

Died: 30 June 2002 in Paris, France

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**Claude Berge**'s parents were André Berge and Geneviève Fourcade. André Berge (1902-1995) was a physician and psychoanalyst who, in addition to his professional work, had published several novels. He was the son of the René Berge, a mining engineer, and Antoinette Faure. Félix François Faure (1841-1899) was Antoinette Faure's father; he was President of France from 1895 to 1899. André Berge married Geneviève in 1924 and Claude, the subject of this biography, was the second of their six children. His five siblings were Nicole (the eldest), Antoine, Philippe, Edith and Patrick. Claude attended the École des Roches near Verneuil-sur-Avre about 110 km west of Paris. This famous private school, founded by the sociologist Edmond Demolins in 1899, attracted students from all over France to its innovative educational programme. At this stage in his life Claude was unsure about the topic in which he should specialise. He said in later life [5]:-

His love of literature and other non-mathematical subjects never left him and we shall discuss below how they played a large role in his life. However, he decided to study mathematics at the University of Paris. After the award of his first degree, he continued to undertake research for his doctorate, advised by André Lichnerowicz. He began publishing mathematics papers in 1950. In that year two of his papers appeared, the short paperI wasn't quite sure that I wanted to do mathematics. There was often a greater urge to study literature.

*Sur l'isovalence et la régularité des transformateurs*Ⓣ and the major 30-page paper

*Sur un nouveau calcul symbolique et ses applications*Ⓣ. The symbolic calculus which he discussed in this major paper is a combination of generating functions and Laplace transforms. He then applied this symbolic calculus to combinatorial analysis, Bernoulli numbers, difference equations, differential equations and summability factors. In 1951 he published a further two short papers

*Sur l'inversion des transformateurs*Ⓣ and

*Sur une théorie ensembliste des jeux alternatifs*Ⓣ that announced various results that would be discussed fully in his thesis. He was awarded a doctorate in 1953 for his thesis

*Sur une théorie ensembliste des jeux alternatifs*Ⓣ. In this thesis he examined games where perfect information is available in which, at each move, there are possibly an infinite number of choices. The games are not necessarily finite, with indefinite continuation being allowed. Berge examined properties of such games with a thorough analysis. A 55-page paper based on his thesis, and with the same title, was published in 1953.

Berge married Jane Gentaz (born 7 January 1925) on 29 December 1952; they had one child, Delphine, born on 1 March 1964. In 1952, before the award of his doctorate, Berge was appointed as a research assistant at the Centre National de la Recherche Scientifique. In 1957 he spent time in the United States as a visiting professor at Princeton University. He took part in the Economics Research Project there which was under contract with the Office of Naval Research. While in Princeton he undertook work which was presented in the paper

*Two theorems in graph theory*published in the

*Proceedings*of the National Academy of Sciences of the United States of America. This was one of his first papers on graph theory, his earlier work being on the theory of games and combinatorics. He was writing his famous book

*Théorie des graphes et ses applications*Ⓣ at this time and had just published his book on the theory of games

*Théorie générale des jeux à n personnes*Ⓣ (1957). Returning to France from the United States, Berge took up the position on Director of research at the Centre national de la recherche scientifique. Also in 1957 he was appointed as a professor in the Institute of Statistics of the University of Paris.

*Théorie des graphes et ses applications*Ⓣ was published in 1958 and, remarkably, in the following year his third book

*Espaces topologiques, fonctions multivoques*Ⓣ was published. For a mathematician in their early thirties to publish three major books within as many years is a truly outstanding achievement. In the highly entertaining article [9] Vasek Chvátal writes:-

For reviews of these books by Berge see THIS LINKI used to think of the book on game theory and the book on topology as a couple of false starts from the days before Claude found his true calling in graph theory and combinatorics. A computer search through Mathematical Reviews changed my mind: with each of these two books, Claude left a lasting mark on the subject. I was pleased to learn that the notions of 'Berge equilibrium' and 'Berge strategies' were being studied by game theorists thirty years after the publication of Claude's book; I was pleased to learn that the 'maximum theorem of Berge' and 'Berge upper semicontinuity' were being studied by economists thirty years after the publication of Claude's book on topology. I was pleased to read Mark Walker's words: "The maximum theorem and its generalizations have become one of the most useful tools in economic theory. The theorem - first stated, and proved by C Berge - gives conditions under which ...". It is amusing to speculate that, just as Claude Berge is a combinatorist to many of us combinatorists, he may be a game theorist to some game theorists and he may be a topologist to some economists.

Of course, today it is difficult to realise that 50 years ago graph theory was an unknown topic to most mathematicians. For example, Berge wrote an introductory article on graph theory which appeared in the

*American Mathematical Monthly*in 1964. The article begins by explaining what "graph theory" is [4]:-

Hungarian mathematicians, such as Denes König, had been among the early workers in graph theory. It was natural, therefore, that the first Colloquium on Graph Theory was held in Hungary. It was held 20-22 October 1959 in Dobogoko, in the mountains to the north of Budapest, and organised by the Bolyai Mathematical Society. Among the non-Hungarians at the meeting were Arthur Harold Stone (1916-2000), Cedric Austen Bardell Smith (1917-2002), Bill Tutte and Berge. All had written, or were intending to write, papers in English on graph theory and they met after the Colloquium to agree on terminology. In 1960 Berge attended a conference at the Martin Luther University of Halle-Wittenberg in Halle am Saale, Germany. In a lecture he gave at the conference he made his famous 'strong perfect graph' conjecture but he only published the conjecture three years later in his paperGraph Theory has its origin in a great number of old problems(in the work of Euler, Kirchhoff et al.)and in recent years its range has become vastly greater. We draw a graph each time we want to represent by points a number of individuals, cities, chemical substances, strategic positions, and to join certain pairs of them by arrows, symbolizing a definite relationship. These diagrams are used in different fields under different names: sociograms(psychology), simplexes(topology), electrical circuits(physics), communication or transportation networks(operational research), family trees, etc. Thanks to the process of abstraction, which is so characteristic of twentieth-century mathematics, the properties of all these diagrams have been systematically analysed and a uniform theory has arisen which is applicable to all these fields.

*Some classes of perfect graphs*(1963). Stefan Hougardy, writing in 2006, looks at the impact of this conjecture. It [14]:-

Also in 1960 Berge became a founder member of Oulipo, an organisation which was particularly well suited to combining two of Berge's interests, namely mathematics and literature. The authors of [6] describe Oulipo:-... has had a major impact on the development of graph theory over the last40years. It has led to the definition and study of many new classes of graphs for which the strong perfect graph conjecture has been verified. Powerful concepts and methods have been developed to prove the strong perfect graph conjecture for these special cases. In this paper we survey120of these classes, list their fundamental algorithmic properties and present all known relations between them.

In 1994 Berge wrote a 'mathematical' murder mystery for Oulipo. In this short storyThis group was created on24November1960... Oulipo was a group of writers and mathematicians aiming at exploring in a systematic way formal constraints on the production of literary texts. Although poets had explored for years the formal constraints of versification, the originality of the group was to develop new or rarely studied constraints and to explore them systematically.

*Who killed the Duke of Densmore*(1995), the Duke of Densmore has been murdered by one of his six mistresses, and Holmes and Watson are summoned to solve the case. Watson is sent by Holmes to the Duke's castle but, on his return, the information he conveys to Holmes is very muddled. Holmes uses the information that Watson gives him to construct a graph. He then applies a theorem of György Hajós to the graph which produces the name of the murderer. Other clever contributions of Berge to Oulipo are described in [6].

Another of Berge's interests was in art and sculpture. He described his early sculptures, made in part from stones found in the river Seine, in his book

*Sculptures multipètre*s (1962). Bjarne Toft writes [21]:-

Sculpture was certainly not his only interest in art for he wrote the small bookIn our modern everyday life we are surrounded and bombarded by(too)beautiful, flawless pictures, sculptures and designs. In this stream Claude Berge's sculptures catch our attention, with their authenticity and honesty. They are not pretending to be more than they are. Berge catches again something general and essential, as he did in his mathematics. The sculptures may at first seem just funny, and they certainly have a humorous side. But they have strong personalities in their unique style - you come to like them as you keep looking at them - whether one could live with them if they came alive is another matter!

*L'Art Asmat*Ⓣ describing the art of the Asmat, an ethnic group of people living in Papua New Guinea. Also on this topic he wrote the paper

*Ancestral Sculpture in Asmat Art*(1994).

After this excursion into Berge's interests outside mathematics, we should return to his mathematical contributions. He was Director of the International Computing Centre in Rome (1965-1967). He spent the period 16 February - 30 June 1968 as a visiting professor at Pennsylvania State University. The announcement of his appointment gave details of his previous experience in the United States:-

Berge continued to write important books:In1956-57he was research associate at Princeton University. He has participated in colloquia at Harvard University and the Massachusetts Institute of Technology and has lectured at a number of institutions in the United States, including Stanford University and the University of California, Berkeley.

*Principes de combinatoire*Ⓣ (1968),

*Graphes et hypergraphes*Ⓣ (1970),

*Introduction à la théorie des hypergraphes*Ⓣ (1973),

*Fractional graph theory*(1978),

*Graphes*Ⓣ (1983), and

*Hypergraphes*Ⓣ.

*Combinatoire des ensembles finis*Ⓣ (1987). The first of these,

*Principes de combinatoire*Ⓣ, was essentially lecture notes of a course Berge gave at the Faculty of Science in Paris in 1967-68. It was translated into English and published as

*Principles of combinatorics*in 1971.

*Introduction à la théorie des hypergraphes*Ⓣ was also notes from a lecture course, this time a course given by Berge at the University of Montreal in the summer of 1971.

*Graphes et hypergraphes*Ⓣ was translated into English and published under the title

*Graphs and hypergraphs*in 1973. Bill Jackson, reviewing a later edition writes:-

Berge continued to revise and update the text and the last two books mentioned above, namelyThe book is a classic in its area and has been instrumental in introducing mathematicians to graphs and taking them to the frontiers of current research.

*Graphes*Ⓣ (1983) and

*Hypergraphes*Ⓣ.

*Combinatoire des ensembles finis*Ⓣ (1987), are revised versions of the first half and the second half (respectively) of the original text. Ioan Tomescu, reviewing the hypergraphs book writes:-

Berge has received many honours for his mathematical contributions including the EURO Gold Medal from the Association of European Operational Research Societies in 1989 and the Euler Medal from the Institute of Combinatorics and Its Applications in 1995 (awarded jointly with Ron Graham).The book presents many examples and applications, especially to integer programming and game theory, and every chapter is accompanied by exercises. The style and content of the book reflect throughout the influence of the author's own work and the distinctive flavour of his personal approach to the subject. The book is an excellent source for both student and researcher.

The strong perfect graph conjecture was settled just before Berge's death. Srinivas Bhogle writes [5]:-

Bhogle was one of Berge's research students and recalls some charming personal details of his advisor [5]:-Berge was still alive when the news of the resolution of his more difficult conjecture came in, although he was reportedly very ill. He must have felt happy to see the end of this four-decade long adventure during his lifetime, although I do not think that the proof itself would have given him any great pleasure. Berge was very much a product of the old school and 'clean' and 'pretty' proofs mattered to him almost as much as the proof itself.

Let us end with Pierre Rosenstiehl's summary [19]:-His residence at10rue Galvani in Paris'1718°^{th}arrondissement was like an unkempt museum with his own sculpture competing for space with the many Chinese works of art that he so adored. A visit to the Berge residence would find him curled up on a sofa, smoking his favourite pipe and gazing intently and lovingly at an 'objet d'art'. Being one of Berge's foreign students in France, it was my privilege to be invited to his house for Christmas dinners. I have the most wonderful memories of these dinners. Berge could either be in a good mood or sulking like a spoilt child. On one occasion he was inconsolable when his wife served us red wine that was excessively chilled. "It shouldn't be less thanC", he kept grumbling. Thankfully, his spirits improved a little after he found that the steak was done just right.

Mathematician, combinatorist, sculptor, amateur anthropologist, world traveller, collector of Indonesian art, Oulipist, chess player and player of exotic game pieces, author of manuals, professor renowned for his30-minute lectures,35-year member of the Centre de Mathématique Sociale, CRNS researcher emeritus at the EHESS until June30,2002: such was Claude Berge.

**Article by:** *J J O'Connor* and *E F Robertson*

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