**Crispin Nash-Williams**' father worked at the National Museum of Wales where he was the keeper of Archaeology and he was also a Senior Lecturer in Archaeology at University College Cardiff. Crispin's mother was a classics graduate of Oxford University. Life for the family might have been very different but for the outbreak of World War II which occurred shortly before Crispin was six years old. His father joined the army and Crispin was sent as a boarder to Christ Church Cathedral School, Oxford.

Crispin's mother, together with his younger brother Piers, moved to Swaffham in Kent when she was appointed as a classics teacher at the local grammar school. The family now had no real home since Crispin's mother and younger brother lived at the home of the head of the grammar school during term time and rented a flat in Reading for the school holidays. Later she moved to London, teaching at a girl's grammar school there and renting a flat in Chelsea.

The war was coming to an end in 1945 when Nash-Williams left Christ Church Cathedral School in Oxford and entered Rugby School. There his interest in mathematics was greatly encouraged by the mathematics teacher and after completing his School Certificate in 1946 he concentrated entirely on that subject. Hilton writes in [2] that at Rugby:-

His father returned to his work in Cardiff after his war service but the family could not reunite since paying school fees for their two children meant that both parents needed to work. However [2]:-... he was academically very successful, but was also apparently bullied unmercifully.

In the summer between leaving school and entering Trinity Hall, Cambridge, he lived with a family in Grenoble for three months while he studied French. During his first year at Cambridge Nash-Williams spent a considerable amount of time with the boat club and was cox of a Trinity Hall boat. However from his second year onwards he gave up his other interests to concentrate completely on his studies of mathematics. In this he was outstandingly successful and graduated as Senior Wrangler in 1953.... an ensuing tragedy was that, his father dying at the relatively young age of58, Crispin never really knew him well.

After graduating, Nash-Williams remained at Cambridge where he undertook research under Shaun Wylie and Davis Rees. He was supported by a scholarship and he was then awarded a Visiting Fellowship to Princeton where he studied during 1956-57. Norman Steenrod was a considerable influence on Nash-Williams during this year. When he returned to the UK, Nash-Williams was appointed as Assistant Lecturer in Mathematics at the University of Aberdeen in October 1957. He was still working on his doctoral thesis, but the first two papers which he submitted were not part of this thesis. He submitted *Abelian groups, graphs and generalized knights* and *Random walk and electric currents in networks* to the *Proceeding of the Cambridge Philosophical Society* on the same day. Both were published in 1959. In the first of these papers Nash-Williams considered an infinite chessboard in *a*-dimensional space for some cardinal number *a*. It has the property that every square has only finitely many non-zero coordinates. In the paper necessary and sufficient conditions are given so that a knight can visit each square exactly once in a single infinite sequence of moves. The problem is solve by reformulating it as a question about infinite abelian groups.

In the second of these two papers Nash-Williams considers a recurrent graph, namely one in which if you start at any vertex and move at random to an adjacent vertex then you will return eventually to the starting vertex with probability 1. In the paper Nash-Williams characterises infinite recurrent graphs. D G Kendall writes in a review of the paper that graphs satisfying Nash-Williams' conditions:-

Nash-Williams' doctoral thesis... are currently of considerable interest, and there is a huge class of[such graphs]of practical importance(mostly corresponding to variants of the random walk).[Nash-Williams']principal result is accordingly very valuable ...

*Decomposition of graphs into infinite chains*was submitted to Cambridge University in 1958 and the degree was awarded in the following year. Not only was it a remarkable piece of mathematical work but the thesis was also remarkable for its length being over 500 pages. A number of papers came out of the work of the thesis, the first being

*Decomposition of graphs into closed and endless chains*published in the

*Proceedings of the London Mathematical Society*in 1960. Hilton, in [2], summarises Nash-Williams' mathematical interests:-

[In fact Nash-Williams had a particular liking for infinite graphs as he expressed in the Introduction to the Proceedings of the conference onHe]was especially interested in aspects of graph theory and he may justly be counted among the founders of the subject, and he contributed greatly to its current status as a serious mathematical subject in its own right. Themes running through his papers are Hamiltonian cycles, Eulerian graphs, spanning trees, the Marriage problem, detachments, reconstruction, and infinite graphs.

*Directions in Infinite Graph Theory and Combinatorics:-*

At Aberdeen Nash-Williams was promoted to Senior Lecturer in Mathematics in 1964 then visited the University of Waterloo in Canada as a Visiting Professor in the following year. The Department of Combinatorics was established at Waterloo in 1967 and Nash-Williams left Aberdeen to become one of the founding professors. After five years, during which he helped build a strong group of research students in the department, he returned to Scotland to become Professor of Pure Mathematics at Aberdeen. He attended the third British Combinatorial Conference which was held in Oxford in 1972 and became part of a committee set up at that conference to make such conferences regular events. The Fifth British Combinatorial Conference was held in Aberdeen in 1975 and Nash-Williams wrote in the Preface of the Proceedings of the Conference:-It has been reported that Denes Konig, the author of the classic 'Theorie der endlichen und unendlichen Graphen'(Leipzig,1936), expressed a special liking for infinite graphs, which certainly receive substantial attention in his book. Nevertheless, the majority of combinatorialists seem to have concentrated on finite combinatorics, to the extent that it has almost seemed an eccentricity to think that graphs and other combinatorial structures can be either finite or infinite.

However, there seems to be no logical reason why combinatorial structures should 'usually' be finite, and indeed this would preclude many fascinating avenues of exploration. To a considerable extent, finite and infinite combinatorics are parts of the same subject. Most of the concepts of finite combinatorics and many of its results carry over(sometimes in more than one way)to the infinite case. Results and problems in infinite combinatorics often arise from seeking analogues of corresponding finite results, and sometimes the attempt to do this also leads to new ideas in finite combinatorics.

Nevertheless, infinite combinatorics gas its own distinctive features. Some of its problems, such as certain ones involving ends of graphs, have no meaningful finite analogue but are closely related to other parts of mathematics. Sometimes problems which are difficult for finite structures become trivial or easy in the infinite case because infinite structures allow so much more room for manoeuvre. On the other hand, the passage from finite to infinite structures often introduces new difficulties ... Sometimes these two phenomena occur together, i.e. passing to the infinite case of a problem may reduce some difficulties while introducing others. Increasingly, interactions between infinite combinatorics and mathematical logic are coming to light.

Shortly after the 1975 conference Nash-Williams moved to Reading where he was appointed to the chair of mathematics following Richard Rado retirement. Hilton writes [2]:-The Fifth British Combinatorial Conference at the University of Aberdeen took place during the period14-18July1975inclusive, and included eight invited lectures by Professors C Berge, G A Dirac, P Erdős, F Harary, L Lovász, Richard Rado and R M Wilson and Principal E M Wright. ... Although there had been two earlier conferences on combinatorics in Britain, the hope that British Combinatorial Conferences might become a regular event probably began to take shape at the Oxford Conference in1972, where a small informal committee was created to coordinate plans for future conferences including ones at Aberystwyth in1973and at Aberdeen in1975.

Such care, of course, meant that his lectures were a joy at attend as I [EFR] can indeed relate through personal experience having attended many excellent lectures by Nash-Williams at conferences. Welsh writes in [7]:-The course notes for his courses at Reading could be extremely long and detailed, and the notes on one course 'Introduction to Analysis' in particular provoked a protest at the Staff/Student Committee that the quantity of notes handed out(averaging17closely written pages per lecture)were unreasonable, and no student could have sufficient time to actually read them.

Such attention to detail was also obvious in the surveys which he wrote. One such survey appeared as a two-part paperNash-Williams' lectures were superbly organised. No detail was omitted and yet the effect was one of simplicity, with the main ideas clearly highlighted.

*A glance at graph theory*published in the

*Bulletin of the London Mathematical Society*. This survey was based on lectures which Nash-Williams gave at the Edinburgh Mathematical Colloquium held in St Andrews in 1980. I [EFR] had the privilege of attending these lectures which were extremely successful in meeting Nash-Williams' aim:-

Other earlier surveys included... of developing nontrivial and fairly deep mathematics from a very simple initial concept.

*Infinite graphs - a survey*(1967) which Nash-Williams describes as follows:-

AlsoThis expository article describes work which has been done on various problems involving infinite graphs, mentioning also a few unsolved problems or suggestions for future investigation.

*Hamiltonian circuits*(1975) which he describes in the introduction as follows:-

Nash-Williams did not enjoy administration and in particular being Head of department at Reading was a chore which he did for six years out of a sense of duty rather than for any other reason. In 1996 he retired, somewhat earlier than was necessary since he wanted to devote time to mathematics free from any worries of administration. A conference was held to mark his retirement and the 272 pageA persistent theme in graph theory has been a desire to determine, in some reasonable sense, which graphs have Hamiltonian circuits and which have not, i.e., we want necessary and sufficient conditions for a graph to have a Hamiltonian circuit. Of course, such necessary and sufficient conditions must be of a psychologically satisfactory kind, and we should not, for example, want a theorem which merely said, perhaps in a slightly disguised form, that a graph has a Hamiltonian circuit if and only if it has a Hamiltonian circuit. ... Even if it exists, however, experience suggests that the problem of discovering it might well be of the same order of difficulty as the four colour problem. This situation has, however, not deterred graph-theorists from studying the problem and obtaining some results which, although far from constituting a complete solution, are nevertheless interesting. This paper will review some of these.

*Festschrift for C St J A Nash-Williams*was produced which contains [4] and [5]. Sadly his retirement was short for in the summer of 2000 he fell ill with cancer and, after a major operation, he had to move into a retirement home since he was no longer able to look after himself. He went to a home in Ascot so that he could be near to his brother Piers who was Rector of Ascot.

After his death the 18^{th} British Combinatorial Conference was held in his memory at the University of Sussex, Sussex, from 1 July to 6 July 2001.

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