WebApr 5, 2011 · Theorem 2 (K}onig) Given a rectangular 0 1 matrix M= (a ij) where 1 i mand 1 j n, de ne a \line" of Mto be a row or column of M. Then the minimum number of lines … WebJun 13, 2024 · The transcript used in this video was heavily influenced by Dr. Oscar Levin's free open-access textbook: Discrete Mathematics: An Open Introduction. Please v...
Understanding Hall
http://voutsadakis.com/TEACH/LECTURES/GRAPHS/Chapter8.pdf WebKo¨nig’s theorem for matrices (1931), the Ko¨nig-Egerv´ary theo-rem (1931), Hall’s marriage theorem (1935), the Birkhoff-Von Neumann theorem (1946), Dilworth’s theorem (1950) and the Max Flow-Min Cut theorem (1962). I will attempt to explain each theorem, and give some indications why all are equivalent. home financing with bad credit nebraska
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WebDilworth's Theorem is a result about the width of partially ordered sets. It is equivalent to (and hence can be used to prove) several beautiful theorems in combinatorics, including Hall's marriage theorem. One well-known corollary of Dilworth's theorem is a result of Erdős and Szekeres on sequences of real numbers: every sequence of rs+1 real … In mathematics, Hall's marriage theorem, proved by Philip Hall (1935), is a theorem with two equivalent formulations: The combinatorial formulation deals with a collection of finite sets. It gives a necessary and sufficient condition for being able to select a distinct element from each set.The graph … See more Statement Let $${\displaystyle {\mathcal {F}}}$$ be a family of finite sets. Here, $${\displaystyle {\mathcal {F}}}$$ is itself allowed to be infinite (although the sets in it are not) and to contain the same … See more Let $${\displaystyle G=(X,Y,E)}$$ be a finite bipartite graph with bipartite sets $${\displaystyle X}$$ and $${\displaystyle Y}$$ and edge set $${\displaystyle E}$$. An $${\displaystyle X}$$-perfect matching (also called an $${\displaystyle X}$$-saturating … See more Marshall Hall Jr. variant By examining Philip Hall's original proof carefully, Marshall Hall Jr. (no relation to Philip Hall) was … See more When Hall's condition does not hold, the original theorem tells us only that a perfect matching does not exist, but does not tell what is the largest … See more Hall's theorem can be proved (non-constructively) based on Sperner's lemma. See more This theorem is part of a collection of remarkably powerful theorems in combinatorics, all of which are related to each other in an … See more A fractional matching in a graph is an assignment of non-negative weights to each edge, such that the sum of weights adjacent to each vertex is at most 1. A fractional matching … See more WebOne says that G satisfies the Hall marriage conditions if G satisfies both the left and the right Hall conditions. Theorem H.3.2. Let G =(X,Y,E) be a locally finite bipartite graph. Then the following conditions are equivalent. (a) G satisfies the left (resp. right) Hall condition; (b) G admits a left (resp. right) perfect matching. Proof. home financing for people with bad credit