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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]
The following describes the equitable partition that the hypergraph regularity lemma will induce. A (,,,)-equitable family of partition is a sequence of partitions of 1-edges (vertices), 2-edges (pairs), 3-edges (triples), etc. This is an important distinction from the partition obtained by Szemerédi's regularity lemma, where only vertices are ...
Lemma II (Lovász 1977; published by Joel Spencer [3]) If (+), where e = 2.718... is the base of natural logarithms, then there is a nonzero probability that none of the events occurs. Lemma II today is usually referred to as "Lovász local lemma". Lemma III (Shearer 1985 [4]) If
The exact origins of the LTE lemma are unclear; the result, with its present name and form, has only come into focus within the last 10 to 20 years. [1] However, several key ideas used in its proof were known to Gauss and referenced in his Disquisitiones Arithmeticae. [2]
This is a simple example of double counting, often used when teaching multiplication to young children. In this context, multiplication of natural numbers is introduced as repeated addition, and is then shown to be commutative by counting, in two different ways, a number of items arranged in a rectangular grid.