sperner's lemma: an application of graph theory @let@token...
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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?