Combinatorics (Fall 2010)/Finite set systems: Difference between revisions

From TCS Wiki
Jump to navigation Jump to search
imported>WikiSysop
imported>WikiSysop
Line 4: Line 4:
=== Hall's marriage theorem ===
=== Hall's marriage theorem ===


{{Theorem|Hall's Theorem|
{{Theorem|Hall's Theorem (SDR)|
:The sets <math>S_1,S_2,\ldots,S_m</math> have a system of distinct representatives (SDR) if and only if
:The sets <math>S_1,S_2,\ldots,S_m</math> have a system of distinct representatives (SDR) if and only if
::<math>\left|\bigcup_{i\in I}S_i\right|\ge |I|</math> for all <math>I\subseteq\{1,2,\ldots,m\}</math>.
::<math>\left|\bigcup_{i\in I}S_i\right|\ge |I|</math> for all <math>I\subseteq\{1,2,\ldots,m\}</math>.
}}
{{Theorem|Hall's Theorem (matching in bipartite graph)|
:A bipartite graph <math>G(U,V,E)</math> has a matching of <math>U</math> if and only if
::<math>\left|N(S)\right|\ge |S|</math> for all <math>S\subseteq U</math>.
}}
}}



Revision as of 03:06, 20 October 2010

Systems of Distinct Representatives (SDR)

Hall's marriage theorem

Hall's Theorem (SDR)
The sets [math]\displaystyle{ S_1,S_2,\ldots,S_m }[/math] have a system of distinct representatives (SDR) if and only if
[math]\displaystyle{ \left|\bigcup_{i\in I}S_i\right|\ge |I| }[/math] for all [math]\displaystyle{ I\subseteq\{1,2,\ldots,m\} }[/math].
Hall's Theorem (matching in bipartite graph)
A bipartite graph [math]\displaystyle{ G(U,V,E) }[/math] has a matching of [math]\displaystyle{ U }[/math] if and only if
[math]\displaystyle{ \left|N(S)\right|\ge |S| }[/math] for all [math]\displaystyle{ S\subseteq U }[/math].

Doubly stochastic matrices

Theorem (Birkhoff 1949; von Neumann 1953)
Every doubly stochastic matrix is a convex combination of permutation matrices.

Cuckoo hashing

Min-max theorems

  • König-Egerváry theorem
  • Menger's theorem
  • Dilworth's theorem
Theorem (König 1931; Egerváry 1931)
In any bipartite graph, the size of a maximum matching equals the size of a minimum vertex cover.
Theorem (Menger 1927)
Let [math]\displaystyle{ G }[/math] be a graph and let [math]\displaystyle{ s }[/math] and [math]\displaystyle{ t }[/math] be two vertices of [math]\displaystyle{ G }[/math]. The maximum number of internally disjoint paths from [math]\displaystyle{ s }[/math] to [math]\displaystyle{ t }[/math] equals the minimum number of vertices in a[math]\displaystyle{ s }[/math]-[math]\displaystyle{ t }[/math] separating set.
Theorem (Dilworth 1950)
Suppose that the largest antichain in the poset [math]\displaystyle{ P }[/math] has size [math]\displaystyle{ r }[/math]. Then [math]\displaystyle{ P }[/math] can be partitioned into [math]\displaystyle{ r }[/math] chains.

Chains and antichains

Symmetric chains

Sperner system