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In graph theory, a tournament is a directed graph with exactly one edge between each two vertices, in one of the two possible directions. Equivalently, a tournament is an orientation of an undirected complete graph. (However, as directed graphs, tournaments are not complete: complete directed graphs have two edges, in both directions, between ...
Among directed graphs, the oriented graphs are the ones that have no 2-cycles (that is at most one of (x, y) and (y, x) may be arrows of the graph). [ 1] A tournament is an orientation of a complete graph. A polytree is an orientation of an undirected tree. [ 2] Sumner's conjecture states that every tournament with 2n – 2 vertices contains ...
e. A tournament solution is a function that maps an oriented complete graph to a nonempty subset of its vertices. It can informally be thought of as a way to find the "best" alternatives among all of the alternatives that are "competing" against each other in the tournament. Tournament solutions originate from social choice theory, [1] [2] [3 ...
Calculus. The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at each point in time) with the concept of integrating a function (calculating the area under its graph, or the cumulative effect of small contributions). Roughly speaking, the two operations ...
In the mathematical field of graph theory, a Hamiltonian path (or traceable path) is a path in an undirected or directed graph that visits each vertex exactly once. A Hamiltonian cycle (or Hamiltonian circuit) is a cycle that visits each vertex exactly once. A Hamiltonian path that starts and ends at adjacent vertices can be completed by adding ...
Hajós construction. In graph theory, a branch of mathematics, the Hajós construction is an operation on graphs named after György Hajós ( 1961) that may be used to construct any critical graph or any graph whose chromatic number is at least some given threshold.
In mathematical analysis, Schwarz's theorem (or Clairaut's theorem on equality of mixed partials) [9] named after Alexis Clairaut and Hermann Schwarz, states that for a function : defined on a set , if is a point such that some neighborhood of is contained in and has continuous second partial derivatives on that neighborhood of , then for all i ...
In mathematics, the closed graph theorem may refer to one of several basic results characterizing continuous functions in terms of their graphs. Each gives conditions when functions with closed graphs are necessarily continuous. A T. Tao ’s blog post [1] lists several closed graph theorems throughout mathematics.