In combinatorial mathematics, two Latin squares of the same size (order) are said to be orthogonal if when superimposed the ordered paired entries in the positions are all distinct. A set of Latin squares, all of the same order, all pairs of which are orthogonal is called a set of mutually orthogonal Latin squares. This concept of orthogonality in combinatorics is strongly related to the concept of blocking in statistics, which ensures that independent variables are truly independent with no hidden confounding correlations. "Orthogonal" is thus synonymous with "independent" in that knowing one variable's value gives no further information about another variable's likely value.
In combinatorial mathematics, a Levi graph or incidence graph is a bipartite graph associated with an incidence structure. From a collection of points and lines in an incidence geometry or a projective configuration, we form a graph with one vertex per point, one vertex per line, and an edge for every incidence between a point and a line. They are named for Friedrich Wilhelm Levi, who wrote about them in 1942.
In mathematics, a two-graph is a set of (unordered) triples chosen from a finite vertex setX, such that every (unordered) quadruple from X contains an even number of triples of the two-graph. A regular two-graph has the property that every pair of vertices lies in the same number of triples of the two-graph. Two-graphs have been studied because of their connection with equiangular lines and, for regular two-graphs, strongly regular graphs, and also finite groups because many regular two-graphs have interesting automorphism groups.
Combinatorial design theory is the part of combinatorial mathematics that deals with the existence, construction and properties of systems of finite sets whose arrangements satisfy generalized concepts of balance and/or symmetry. These concepts are not made precise so that a wide range of objects can be thought of as being under the same umbrella. At times this might involve the numerical sizes of set intersections as in block designs, while at other times it could involve the spatial arrangement of entries in an array as in sudoku grids.
Jeffrey Howard Dinitz is an American mathematician who taught combinatorics at the University of Vermont. He is best known for proposing the Dinitz conjecture, which became a major theorem.
In mathematics, a conference matrix (also called a C-matrix) is a square matrix C with 0 on the diagonal and +1 and −1 off the diagonal, such that CTC is a multiple of the identity matrix I. Thus, if the matrix has order n, CTC = (n−1)I. Some authors use a more general definition, which requires there to be a single 0 in each row and column but not necessarily on the diagonal.
In mathematics a regular Hadamard matrix is a Hadamard matrix whose row and column sums are all equal. While the order of a Hadamard matrix must be 1, 2, or a multiple of 4, regular Hadamard matrices carry the further restriction that the order must be a square number. The excess, denoted E(H ), of a Hadamard matrix H of order n is defined to be the sum of the entries of H. The excess satisfies the bound |E(H )| ≤ n3/2. A Hadamard matrix attains this bound if and only if it is regular.
In mathematics, the Turán number T(n,k,r) for r-uniform hypergraphs of order n is the smallest number of r-edges such that every induced subgraph on k vertices contains an edge. This number was determined for r = 2 by Turán (1941), and the problem for general r was introduced in Turán (1961). The paper (Sidorenko 1995) gives a survey of Turán numbers.
William Lawrence Kocay is a Canadian professor at the department of computer science at St. Paul's College of the University of Manitoba and a graph theorist. He is known for his work in graph algorithms and the reconstruction conjecture and is affectionately referred to as "Wild Bill" by his students. Bill Kocay is a former managing editor of Ars Combinatoria, a Canadian journal of combinatorial mathematics, is a founding fellow of the Institute of Combinatorics and its Applications.
Ernest Tilden Parker (1926–1991) was a professor emeritus of the University of Illinois Urbana-Champaign. He is notable for his breakthrough work along with R. C. Bose and S. S. Shrikhande in their disproof of the famous conjecture made by Leonhard Euler dated 1782 that there do not exist two mutually orthogonal latin squares of order for every . He was at that time employed in the UNIVAC division of Remington Rand, but he subsequently joined the mathematics faculty at t University of Illinois. In 1968, he and a Ph.D. student, K. B. Reid, disproved a conjecture on tournaments by Paul Erdős and Leo Moser.
In combinatorial mathematics, a Latin rectangle is an r × n matrix, using n symbols, usually the numbers 1, 2, 3, ..., n or 0, 1, ..., n − 1 as its entries, with no number occurring more than once in any row or column.
Charles Joseph Colbourn is a Canadian computer scientist and mathematician, whose research concerns graph algorithms, combinatorial designs, and their applications. From 1996 to 2001 he was the Dorothean Professor of Computer Science at the University of Vermont; since then he has been a professor of Computer Science and Engineering at Arizona State University.
In graph-theoretic mathematics, a biregular graph or semiregular bipartite graph is a bipartite graph for which every two vertices on the same side of the given bipartition have the same degree as each other. If the degree of the vertices in is and the degree of the vertices in is , then the graph is said to be -biregular.
Miklós Bóna is an American mathematician of Hungarian origin.
Choi Seok-jeong was a Korean politician and mathematician in the Joseon period of Korea.
The Combinatorial Mathematics Society of Australasia (CMSA) is a professional society of mathematicians working in the field of combinatorics. It is the primary combinatorics society for Australasia, consisting of Australia, New Zealand and neighbouring countries. The CMSA existed as an informal group from 1972 until formal establishment in 1978. It became an incorporated association in 1996, and as of 2017, it has over 280 members including 110 life members.
Anne Penfold Street (1932–2016) was one of Australia's leading mathematicians, specialising in combinatorics. She was the third woman to become a mathematics professor in Australia, following Hanna Neumann and Cheryl Praeger. She was the author of several textbooks, and her work on sum-free sets became a standard reference for its subject matter. She helped found several important organizations in combinatorics, developed a researcher network, and supported young students with interest in mathematics.
Catherine Greenhill is an Australian mathematician known for her research on random graphs, combinatorial enumeration and Markov chains. She is a professor of mathematics in the School of Mathematics and Statistics at the University of New South Wales, and an editor-in-chief of the Electronic Journal of Combinatorics.
Jeanette Claire McLeod is a New Zealand mathematician specialising in combinatorics, including the theories of Latin squares and random graphs. She is a senior lecturer in the School of Mathematics and Statistics at the University of Canterbury, a principal investigator for Te Pūnaha Matatini, a Centre of Research Excellence associated with the University of Auckland, an honorary senior lecturer at the Australian National University, and the president for three terms from 2018 to 2020 of the Combinatorial Mathematics Society of Australasia.
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.
