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For example, if there is a party of people who shake hands, the number of people who shake an odd number of other people's hands is even. [1] The handshaking lemma is a consequence of the degree sum formula, also sometimes called the handshaking lemma, [2] according to which the sum of the degrees (the numbers of times each vertex is touched ...
A trivial example. In mathematics, the mountain climbing problem is a mathematical problem that considers a two-dimensional mountain range (represented as a continuous function), and asks whether it is possible for two mountain climbers starting at sea level on the left and right sides of the mountain to meet at the summit, while maintaining equal altitudes at all times.
Burnside's lemma also known as the Cauchy–Frobenius lemma; Frattini's lemma (finite groups) Goursat's lemma; Mautner's lemma (representation theory) Ping-pong lemma (geometric group theory) Schreier's subgroup lemma; Schur's lemma (representation theory) Zassenhaus lemma
A graph with a loop having vertices labeled by degree. In graph theory, the degree (or valency) of a vertex of a graph is the number of edges that are incident to the vertex; in a multigraph, a loop contributes 2 to a vertex's degree, for the two ends of the edge. [1]
Some authors use the term Alexander trick for the statement that every homeomorphism of can be extended to a homeomorphism of the entire ball .. However, this is much easier to prove than the result discussed above: it is called radial extension (or coning) and is also true piecewise-linearly, but not smoothly.
This theorem states that every planar graph has a graph coloring of its the vertices with four colors, but Tait showed how to convert 4-vertex-colorings of maximal planar graphs into 3-edge-colorings of their dual graphs, which are cubic and planar, and vice versa. A planar snark would therefore necessarily be dual to a counterexample to the ...
As is often the case with probabilistic arguments, this theorem is nonconstructive and gives no method of determining an explicit element of the probability space in which no event occurs. However, algorithmic versions of the local lemma with stronger preconditions are also known (Beck 1991; Czumaj and Scheideler 2000).
Similarly, the hypergraph counting lemma is a generalization of the graph counting lemma that estimates number of copies of a fixed graph as a subgraph of a larger graph. There are several distinct formulations of the method, all of which imply the hypergraph removal lemma and a number of other powerful results, such as Szemerédi's theorem ...