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, from evolutionary biology to computer science, etc.
Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. In contrast to real numbers that have the property of varying "smoothly", the objects studied in discrete mathematics – such as integers, graphs, and statements in logic – do not vary smoothly in this way, but have distinct, separated values. Discrete mathematics therefore excludes topics in "continuous mathematics" such as calculus or Euclidean geometry. Discrete objects can often be enumerated by integers. More formally, discrete mathematics has been characterized as the branch of mathematics dealing with countable sets. However, there is no exact definition of the term "discrete mathematics." Indeed, discrete mathematics is described less by what is included than by what is excluded: continuously varying quantities and related notions.
Ronald Lewis Graham was an American mathematician credited by the American Mathematical Society as "one of the principal architects of the rapid development worldwide of discrete mathematics in recent years". He was president of both the American Mathematical Society and the Mathematical Association of America, and his honors included the Leroy P. Steele Prize for lifetime achievement and election to the National Academy of Sciences.
Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. In contrast to real numbers that have the property of varying "smoothly", the objects studied in discrete mathematics – such as integers, graphs, and statements in logic – do not vary smoothly in this way, but have distinct, separated values. Discrete mathematics therefore excludes topics in "continuous mathematics" such as calculus and analysis.
Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects. Most questions in discrete geometry involve finite or discrete sets of basic geometric objects, such as points, lines, planes, circles, spheres, polygons, and so forth. The subject focuses on the combinatorial properties of these objects, such as how they intersect one another, or how they may be arranged to cover a larger object.
In the mathematical field of graph theory, a spanning treeT of an undirected graph G is a subgraph that is a tree which includes all of the vertices of G. In general, a graph may have several spanning trees, but a graph that is not connected will not contain a spanning tree. If all of the edges of G are also edges of a spanning tree T of G, then G is a tree and is identical to T.
In the mathematical field of graph theory, a complete bipartite graph or biclique is a special kind of bipartite graph where every vertex of the first set is connected to every vertex of the second set.
Paul Erdős was a renowned Hungarian mathematician. He was one of the most prolific mathematicians and producers of mathematical conjectures of the 20th century. He was known both for his social practice of mathematics and for his eccentric lifestyle. He devoted his waking hours to mathematics, even into his later years—indeed, his death came only hours after he solved a geometry problem at a conference in Warsaw.
Béla Bollobás FRS is a Hungarian-born British mathematician who has worked in various areas of mathematics, including functional analysis, combinatorics, graph theory, and percolation. He was strongly influenced by Paul Erdős since the age of 14.
In graph theory, Schnyder's theorem is a characterization of planar graphs in terms of the order dimension of their incidence posets. It is named after Walter Schnyder, who published its proof in 1989.
Imre Bennett Leader is a British mathematician and Othello player. He is Professor of Pure Mathematics, specifically combinatorics, at the University of Cambridge.
In order theory, a branch of mathematics, a linear extension of a partial order is a total order that is compatible with the partial order. As a classic example, the lexicographic order of totally ordered sets is a linear extension of their product order.
Combinatorica is an international journal of mathematics, publishing papers in the fields of combinatorics and computer science. It started in 1981, with László Babai and László Lovász as the editors-in-chief with Paul Erdős as honorary editor-in-chief. The current editors-in-chief are László Babai, László Lovász, and Alexander Schrijver. The advisory board consists of Ronald Graham, András Hajnal, Gyula O. H. Katona, Miklós Simonovits, and Vera Sós. It is published by the János Bolyai Mathematical Society and Springer Verlag.
Norman Linstead Biggs is a leading British mathematician focusing on discrete mathematics and in particular algebraic combinatorics.
In mathematics, an incidence poset or incidence order is a type of partially ordered set that represents the incidence relation between vertices and edges of an undirected graph. The incidence poset of a graph G has an element for each vertex or edge in G; in this poset, there is an order relation x ≤ y if and only if either x = y or x is a vertex, y is an edge, and x is an endpoint of y.
Yoshiharu Kohayakawa is a Japanese-Brazilian mathematician working on discrete mathematics and probability theory. He is known for his work on Szemerédi's regularity lemma, which he extended to sparser graphs.
Ismat Beg, FPAS, FIMA, is a Pakistani mathematician and researcher. Beg is Professor at the Lahore School of Economics, Higher Education Commission Distinguished National Professor and an honorary full professor at the Mathematics Division of the Institute for Basic Research, Florida, US.
Ilona Palásti (1924–1991) was a Hungarian mathematician who worked at the Alfréd Rényi Institute of Mathematics. She is known for her research in discrete geometry, geometric probability, and the theory of random graphs. With Alfréd Rényi and others, she was considered to be one of the members of the Hungarian School of Probability.
William G. Brown is a Canadian mathematician specializing in graph theory. He is a professor emeritus of mathematics at McGill University.
József Balogh is a Hungarian mathematician, specializing in graph theory and combinatorics.
