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As well as applying differencing to the full 5-bit characters of the ITA2 code, Tutte applied it to the individual impulses bits.
After explaining his findings to Max NewmanNewman was given the job of developing an automated approach to comparing ciphertext and key to look for departures from randomness. The first machine was dubbed Heath Robinsonbut the much faster Colossus computerdeveloped by Tommy Flowers and using algorithms written by Tutte and his colleagues, soon took over for breaking codes.
In lateTutte resumed his studies at Cambridge, now as a graduate student in mathematics. He published some work begun earlier, one a now famous paper that characterises which graphs have a perfect matching, and another that constructs a non-Hamiltonian graph.
He went on to create a ground-breaking PhD thesis, An algebraic theory of graphs, about the subject later known as matroid theory. Inhe moved to the University of Waterloo in WaterlooOntario, where he stayed for the rest of his academic career. He officially retired inbut remained active as an emeritus professor.
Tutte was instrumental in helping Crm thesis 2012 found the Department of Combinatorics and Optimization at the University of Waterloo. His mathematical career concentrated on combinatoricsespecially graph theorywhich he is credited as having helped create in its modern form, and matroid theoryto which he made profound contributions; one colleague described him as "the leading mathematician in combinatorics for three decades".
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He was editor in chief of the Journal of Combinatorial Theory until retiring from Waterloo in Research contributions[ edit ] Tutte's work in graph theory includes the structure of cycle spaces and cut spacesthe size of maximum matchings and existence of k-factors in graphs, and Hamiltonian and non-Hamiltonian graphs.
The eventual proof of the four colour theorem made use of his earlier work.
The graph polynomial he called the "dichromate" has become famous and influential under the name of the Tutte polynomial and serves as the prototype of combinatorial invariants that are universal for all invariants that satisfy a specified reduction law.
The first major advances in matroid theory were made by Tutte in his Cambridge PhD thesis which formed the basis of an important sequence of papers published over the next two decades.
Tutte's work in graph theory and matroid theory has been profoundly influential on the development of both the content and direction of these two fields. In addition, Tutte developed an algorithm for determining whether a given binary matroid is a graphic matroid.
The algorithm makes use of the fact that a planar graph is simply a graph whose circuit-matroid, the dual of its bond-matroidis graphic.
Using this fact, Tutte developed an alternative proof to show that every Kuratowski graph is non-planar by showing that K5 and K3,3 each have three distinct peripheral cycles with a common edge.
In addition to using peripheral cycles to prove that the Kuratowski graphs are non-planar, Tutte proved that every simple 3-connected graph can be drawn with all its faces convex, and devised an algorithm which constructs the plane drawing by solving a linear system.
The resulting drawing is known as the Tutte embedding. Tutte's algorithm makes use of the barycentric mappings of the peripheral circuits of a simple 3-connected graph.
One of the reasons for which Tutte's embedding is popular is that the necessary computations that are carried out by his algorithms are simple and guarantee a one-to-one correspondence of a graph and its embedding onto the Euclidean planewhich is of importance when parameterising a three-dimensional mesh to the plane in geometric modelling.
This work involved some highly innovative techniques of his own invention, requiring considerable manipulative dexterity in handling power series whose coefficients count appropriate kinds of graphs and the functions arising as their sums, as well as geometrical dexterity in extracting these power series from the graph-theoretic situation.
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