What is Sperner's Lemma and how does it relate to Graph Theory? [Discrete Mathematics]

What is Sperner's Lemma? This video will explain Sperner's Lemma and its relation to graph theory.

Sperner's Lemma is an important combinatorial result that has applications to many area of mathematics. It has been used in root-finding algorithms and computing fixed points. Sperner's Lemma relates to graph colorings, triangulations (the subdivision of shapes into triangles) and simplices, or triangles generalized to any dimension. In this video, we will focus primarily on the 2-dimensional case of Sperner's Lemma, but it has been generalized to triangles of any dimension. Sperner's Lemma states that if you start with a triangulation of a triangle and then color its vertices according to a few rules (called a Sperner coloring), then some triangle within the triangulation will be properly colored, or in other words, each vertex of that triangle will have a unique color.

Here are some links for more information:
http://discretemath.imp.fu-berlin.de/DMII-2019-20/Sperner.pdf
https://en.wikipedia.org/wiki/Sperner%27s_lemma
https://www.theoremoftheday.org/Topology/Sperner/TotDSperner.pdf
http://math.mit.edu/~fox/MAT307-lecture03.pdf
https://twocents.lifehacker.com/use-sperner-s-lemma-to-split-rent-fairly-1585414761

--------------------------------- Recommended Books ---------------------------------
********************************* Hypergraph Theory *********************************
"Hypergraph Theory: An Introduction": https://amzn.to/43bC8h1

"Introduction to Graph and Hypergraph Theory": https://amzn.to/3Ij9Poz

********************************* Graph Theory *********************************
"Introduction to Graph Theory (Trudeau)": https://amzn.to/43e2gHR

"Graph Theory (Diestel)": https://amzn.to/3OuLxfw

"The Fascinating World of Graph Theory": https://amzn.to/3pSUFQB

********************************* Other Mathematics *********************************
Discrete Mathematics with Applications (Epp): https://amzn.to/3MBXzkL

A Book of Abstract Algebra (Pinter): https://amzn.to/3On2jgp

Language, Proof and Logic: https://amzn.to/3Oi68n6

Linear Algebra and Its Applications: https://amzn.to/3MhvM91

All the Math You Missed: https://amzn.to/42FqOK5

These are my Amazon Affiliate links. As an Amazon Associate I may earn commisions for purchases made through the links above.

Видео What is Sperner's Lemma and how does it relate to Graph Theory? [Discrete Mathematics] канала Vital Sine