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 applications ranging from logic to statistical physics and from evolutionary biology to computer science.
William Thomas TutteOC FRS FRSC was an English and Canadian codebreaker and mathematician. During the Second World War, he made a brilliant and fundamental advance in cryptanalysis of the Lorenz cipher, a major Nazi German cipher system which was used for top-secret communications within the Wehrmacht High Command. The high-level, strategic nature of the intelligence obtained from Tutte's crucial breakthrough, in the bulk decrypting of Lorenz-enciphered messages specifically, contributed greatly, and perhaps even decisively, to the defeat of Nazi Germany. He also had a number of significant mathematical accomplishments, including foundation work in the fields of graph theory and matroid theory.
In combinatorics, a branch of mathematics, a matroid is a structure that abstracts and generalizes the notion of linear independence in vector spaces. There are many equivalent ways to define a matroid axiomatically, the most significant being in terms of
Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures.
In graph theory, an undirected graph H is called a minor of the graph G if H can be formed from G by deleting edges, vertices and by contracting edges.
In the mathematical field of graph theory, the Heawood graph is an undirected graph with 14 vertices and 21 edges, named after Percy John Heawood.
The Fulkerson Prize for outstanding papers in the area of discrete mathematics is sponsored jointly by the Mathematical Optimization Society (MOS) and the American Mathematical Society (AMS). Up to three awards of $1,500 each are presented at each (triennial) International Symposium of the MOS. Originally, the prizes were paid out of a memorial fund administered by the AMS that was established by friends of the late Delbert Ray Fulkerson to encourage mathematical excellence in the fields of research exemplified by his work. The prizes are now funded by an endowment administered by MPS.
Paul D. Seymour is a British mathematician known for his work in discrete mathematics, especially graph theory. He was responsible for important progress on regular matroids and totally unimodular matrices, the four colour theorem, linkless embeddings, graph minors and structure, the perfect graph conjecture, the Hadwiger conjecture, claw-free graphs, χ-boundedness, and the Erdős–Hajnal conjecture. Many of his recent papers are available from his website.
Jack R. Edmonds is an American-born and educated computer scientist and mathematician who lived and worked in Canada for much of his life. He has made fundamental contributions to the fields of combinatorial optimization, polyhedral combinatorics, discrete mathematics and the theory of computing. He was the recipient of the 1985 John von Neumann Theory Prize.
In graph theory, a mathematical discipline, a factor-critical graph is a graph with n vertices in which every subgraph of n − 1 vertices has a perfect matching.
The Journal of Combinatorial Theory, Series A and Series B, are mathematical journals specializing in combinatorics and related areas. They are published by Elsevier. Series A is concerned primarily with structures, designs, and applications of combinatorics. Series B is concerned primarily with graph and matroid theory. The two series are two of the leading journals in the field and are widely known as JCTA and JCTB.
In graph theory, a branch-decomposition of an undirected graph G is a hierarchical clustering of the edges of G, represented by an unrooted binary tree T with the edges of G as its leaves. Removing any edge from T partitions the edges of G into two subgraphs, and the width of the decomposition is the maximum number of shared vertices of any pair of subgraphs formed in this way. The branchwidth of G is the minimum width of any branch-decomposition of G.
John Adrian Bondy is a retired English mathematician, known for his work in combinatorics and graph theory.
In the mathematical field of graph theory, the Meredith graph is a 4-regular undirected graph with 70 vertices and 140 edges discovered by Guy H. J. Meredith in 1973.
Jim Geelen is a professor at the Department of Combinatorics and Optimization in the faculty of mathematics at the University of Waterloo, where he holds the Canada Research Chair in Combinatorial optimization. He is known for his work on Matroid theory and the extension of the Graph Minors Project to representable matroids. In 2003, he won the Fulkerson Prize with his co-authors A. M. H. Gerards, and A. Kapoor for their research on Rota's excluded minors conjecture. In 2006, he won the Coxeter–James Prize presented by the Canadian Mathematical Society.
In matroid theory, a field within mathematics, a gammoid is a certain kind of matroid, describing sets of vertices that can be reached by vertex-disjoint paths in a directed graph.
In the mathematical theory of matroids, a minor of a matroid M is another matroid N that is obtained from M by a sequence of restriction and contraction operations. Matroid minors are closely related to graph minors, and the restriction and contraction operations by which they are formed correspond to edge deletion and edge contraction operations in graphs. The theory of matroid minors leads to structural decompositions of matroids, and characterizations of matroid families by forbidden minors, analogous to the corresponding theory in graphs.
Rota's excluded minors conjecture is one of a number of conjectures made by the mathematician Gian-Carlo Rota. It is considered an important problem by some members of the structural combinatorics community. Rota conjectured in 1971 that, for every finite field, the family of matroids that can be represented over that field has only finitely many excluded minors. A proof of the conjecture has been announced by Geelen, Gerards, and Whittle.
In graph theory, a branch of mathematics, the kth powerGk of an undirected graph G is another graph that has the same set of vertices, but in which two vertices are adjacent when their distance in G is at most k. Powers of graphs are referred to using terminology similar to that of exponentiation of numbers: G2 is called the square of G, G3 is called the cube of G, etc.
David Ronald Wood is a Professor in the School of Mathematics at Monash University in Melbourne, Australia. His research area is discrete mathematics and theoretical computer science, especially structural graph theory, extremal graph theory, geometric graph theory, graph colouring, graph drawing, and combinatorial geometry.
