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In graph theory, the handshaking lemma is the statement that, in every finite undirected graph, the number of vertices that touch an odd number of edges is even. 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 ]
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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 ...
Abhyankar's lemma; Aubin–Lions lemma; Bergman's diamond lemma; Fitting lemma; Injective test lemma; Hua's lemma (exponential sums) Krull's separation lemma; Schanuel's lemma (projective modules) Schwartz–Zippel lemma; Shapiro's lemma; Stewart–Walker lemma ; Whitehead's lemma (Lie algebras) Zariski's lemma
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Another related result is the so-called counterpart of the Borel–Cantelli lemma. It is a counterpart of the Lemma in the sense that it gives a necessary and sufficient condition for the limsup to be 1 by replacing the independence assumption by the completely different assumption that ( A n ) {\displaystyle (A_{n})} is monotone increasing for ...
In this example, where n=2, the red 1-simplex has vertices which are labelled by the same number with opposite signs. Tucker's lemma states that for such a triangulation at least one such 1-simplex must exist. In mathematics, Tucker's lemma is a combinatorial analog of the Borsuk–Ulam theorem, named after Albert W. Tucker.
In mathematics, the Johnson–Lindenstrauss lemma is a result named after William B. Johnson and Joram Lindenstrauss concerning low-distortion embeddings of points from high-dimensional into low-dimensional Euclidean space.