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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 ...
6.1 Euclidean geometry. 6.2 Hyperbolic geometry. 6.3 Metric spaces. ... Handshaking lemma; Kelly's lemma; Kőnig's lemma; Szemerédi regularity lemma; Order theory.
From the handshaking lemma, a k-regular graph with odd k has an even number of vertices. A theorem by Nash-Williams says that every k ‑regular graph on 2k + 1 vertices has a Hamiltonian cycle. Let A be the adjacency matrix of a graph. Then the graph is regular if and only if = (, …,) is an eigenvector of A. [2]
In more colloquial terms, in a party of people some of whom shake hands, an even number of people must have shaken an odd number of other people's hands; for this reason, the result is known as the handshaking lemma. To prove this by double counting, let () be the degree of vertex . The number of vertex-edge incidences in the graph may be ...
The formula implies that in any undirected graph, the number of vertices with odd degree is even. This statement (as well as the degree sum formula) is known as the handshaking lemma. The latter name comes from a popular mathematical problem, which is to prove that in any group of people, the number of people who have shaken hands with an odd ...
In mathematics and other fields, [a] a lemma (pl.: lemmas or lemmata) is a generally minor, proven proposition which is used to prove a larger statement. For that reason, it is also known as a "helping theorem " or an "auxiliary theorem".
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 ...
The Erdős–Gallai theorem is a result in graph theory, a branch of combinatorial mathematics. It provides one of two known approaches to solving the graph realization problem , i.e. it gives a necessary and sufficient condition for a finite sequence of natural numbers to be the degree sequence of a simple graph .