Anthony J. W. Hilton (born 4 April 1941) is a British mathematician specializing in

combinatorics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many appl ...

and

graph theory In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conne ...

. His current positions are as

emeritus ''Emeritus'' (; female: ''emerita'') is an adjective used to designate a retired chair, professor, pastor, bishop, pope, director, president, prime minister, rabbi, emperor, or other person who has been "permitted to retain as an honorary title ...

professor of Combinatorial Mathematics at the University of Reading and professorial research fellow at Queen Mary College, University of London.

Education

From 1951 to 1959, he attended the

Bedford School :''Bedford School is not to be confused with Bedford Girls' School, Bedford High School, Bedford Modern School, Old Bedford School in Bedford, Texas or Bedford Academy in Bedford, Nova Scotia.'' Bedford School is a public school (English indep ...

Bedford Bedford is a market town in Bedfordshire, England. At the 2011 Census, the population of the Bedford built-up area (including Biddenham and Kempston) was 106,940, making it the second-largest settlement in Bedfordshire, behind Luton, whilst ...

Bedfordshire Bedfordshire (; abbreviated Beds) is a ceremonial county in the East of England. The county has been administered by three unitary authorities, Borough of Bedford, Central Bedfordshire and Borough of Luton, since Bedfordshire County Council wa ...

England England is a country that is part of the United Kingdom. It shares land borders with Wales to its west and Scotland to its north. The Irish Sea lies northwest and the Celtic Sea to the southwest. It is separated from continental Europe b ...

. From there he attended

Reading University The University of Reading is a public university in Reading, Berkshire, England. It was founded in 1892 as University College, Reading, a University of Oxford extension college. The institution received the power to grant its own degrees in 192 ...

, where he earned a bachelor's degree in 1963 and was awarded a PhD in 1967.Hilton, Anthony
Personal Homepage
/ref> His dissertation was "Representation Theorems for Integers and Real Numbers" under his advisor David E. Daykin.Anthony Hilton
The Mathematics Genealogy Project

Work

Much of his work has been done in pioneering techniques in graph theory. He has discovered many results involving

Latin square In combinatorics and in experimental design, a Latin square is an ''n'' × ''n'' array filled with ''n'' different symbols, each occurring exactly once in each row and exactly once in each column. An example of a 3×3 Latin sq ...

s, including, which states that "if

n-1

cells of an

n\times n

matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...

are preassigned with no element repeated in any row or column then the remaining

n^2-n+1

cells can be filled so as to produce a Latin square." Another noteworthy result states that given a ''k''-

regular graph In graph theory, a regular graph is a graph where each vertex has the same number of neighbors; i.e. every vertex has the same degree or valency. A regular directed graph must also satisfy the stronger condition that the indegree and outdegree o ...

with

2n

vertices, if

k \geq 12n/7

then it is 1-factorizable. Chetwynd, A. G.; Hilton, A. J. W. (1985), "Regular graphs of high degree are 1-factorizable", Proceedings of the London Mathematical Society 50 (2): 193–206
doi:10.1112/plms/s3-50.2.193
In 1998, he was awarded the

Euler Medal The Institute of Combinatorics and its Applications (ICA) is an international scientific organization formed in 1990 to increase the visibility and influence of the combinatorial community. In pursuit of this goal, the ICA sponsors conferences, ...

for "a distinguished career in the work he has produced, the people he has trained, and his leadership in the development of combinatorics in Britain." Among the specific things cited for are the creation of two new techniques for solving long standing problems. Through the use of

edge coloring In graph theory, an edge coloring of a graph is an assignment of "colors" to the edges of the graph so that no two incident edges have the same color. For example, the figure to the right shows an edge coloring of a graph by the colors red, blue ...

s in the context of embedding graphs, he was able to settle the Evan's conjectureAnderson; Hilton (1980)
"Thank Evans!"
Proc. London Math. Soc., s3–47 (3) 507–522. and the Lindner conjecture. Through the use of

graph amalgamation In graph theory, a graph amalgamation is a relationship between two graphs (one graph is an amalgamation of another). Similar relationships include subgraphs and minors. Amalgamations can provide a way to reduce a graph to a simpler graph while k ...

s he was able to show many results, including a method for enumerating

Hamiltonian decomposition In graph theory, a branch of mathematics, a Hamiltonian decomposition of a given graph is a partition of the edges of the graph into Hamiltonian cycles. Hamiltonian decompositions have been studied both for undirected graphs and for directed graphs ...

s as well as a

conjecture In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 19 ...

about embedding partial triple systems.Hilton; Roger (1990)
Edge-Colouring Graphs and Embedding Partial Triple Systems of Even Index
NATO ASI Series, Springer Netherlands, 301 pp 101-112

References

{{DEFAULTSORT:Hilton, Anthony 1941 births Living people 20th-century British mathematicians 21st-century British mathematicians Graph theorists