Sperner’s Lemma: An Application of GraphTheory

AMS 550.472/672: Graph Theory

Spring 2016

Johns Hopkins University

A Problem on triangles

I Take any planetriangle

A Problem on triangles

I Take any planetriangle.

I Mark any finitesubset of points,including corners.

A Problem on triangles

I Take any planetriangle.

I Mark any finitesubset of points,including corners.

I Break up intosmaller triangles(any way you like).

A Problem on triangles

Color points with 3colors using two rules:

A Problem on triangles

Color points with 3colors using two rules:

1. Corners getdifferent colors

A Problem on triangles

Color points with 3colors using two rules:

1. Corners getdifferent colors

2. Edge gets colors ofits endpoints

A Problem on triangles

Color points with 3colors using two rules:

1. Corners getdifferent colors

2. Edge gets colors ofits endpoints

Then we have a“multi-colored”triangle.

Simple observation about line segments

I Start with any line segment.

Simple observation about line segments

I Start with any line segment.

I Mark any subset of points on the line segment which includeend points.

Simple observation about line segments

I Start with any line segment.

I Mark any subset of points on the line segment which includeend points.

I Color points using two colors such that end points getdifferent colors.

Then, we have an odd number of “multi-colored” segments.

Graph theory

Vertices + Edges

# of Tokens =

Sum of degrees

# of Tokens =

2*(# of edges)

I Edge-degree of vertex := # of edges incident on it

I THEOREM Sum of the degrees = 2*(# of edges)

I COROLLARY Number of odd degree vertices is even

Proof of Sperner’s Lemma

We create a graph outof the triangles.

Proof of Sperner’s Lemma

We create a graph outof the triangles.

I Put a vertex foreach smalltriangle.

Proof of Sperner’s Lemma

We create a graph outof the triangles.

I Connect vertices⇔ correspondingtriangles sharemulti-colorededge.

Proof of Sperner’s Lemma

We create a graph outof the triangles.

I Put extra vertexfor “outside”.

Proof of Sperner’s Lemma

We create a graph outof the triangles.

I Put edges between“outside” vertexand inner vertex ifinner triangle hasmulti-coloredboundary edge.

Proof of Sperner’s Lemma

We create a graph outof the triangles.

I “Outside” vertexhas odd degree byline segmentobservation.

Proof of Sperner’s Lemma

We create a graph outof the triangles.

I “Outside” vertexhas odd degree byline segmentobservation.

I No degree 1 innervertex.

Proof of Sperner’s Lemma

We create a graph outof the triangles.

I “Outside” vertexhas odd degree byline segmentobservation.

I No degree 1 innervertex.

By degree-sum formula, there are an odd number (therefore, atleast 1) of degree 3 inner vertices = “completely” coloredtriangles.

Questions?