Sperner's Lemma: An Application of Graph Theory @let@token ...abasu9/AMS_550-472-672/sperner.pdf ·...
Transcript of Sperner's Lemma: An Application of Graph Theory @let@token ...abasu9/AMS_550-472-672/sperner.pdf ·...
![Page 1: Sperner's Lemma: An Application of Graph Theory @let@token ...abasu9/AMS_550-472-672/sperner.pdf · Proof of Sperner’s Lemma We create a graph out of the triangles. I \Outside"](https://reader034.fdocuments.us/reader034/viewer/2022050504/5f963b15c5d236713860cea9/html5/thumbnails/1.jpg)
Sperner’s Lemma: An Application of GraphTheory
AMS 550.472/672: Graph Theory
Spring 2016
Johns Hopkins University
![Page 2: Sperner's Lemma: An Application of Graph Theory @let@token ...abasu9/AMS_550-472-672/sperner.pdf · Proof of Sperner’s Lemma We create a graph out of the triangles. I \Outside"](https://reader034.fdocuments.us/reader034/viewer/2022050504/5f963b15c5d236713860cea9/html5/thumbnails/2.jpg)
A Problem on triangles
I Take any planetriangle
![Page 3: Sperner's Lemma: An Application of Graph Theory @let@token ...abasu9/AMS_550-472-672/sperner.pdf · Proof of Sperner’s Lemma We create a graph out of the triangles. I \Outside"](https://reader034.fdocuments.us/reader034/viewer/2022050504/5f963b15c5d236713860cea9/html5/thumbnails/3.jpg)
A Problem on triangles
I Take any planetriangle.
I Mark any finitesubset of points,including corners.
![Page 4: Sperner's Lemma: An Application of Graph Theory @let@token ...abasu9/AMS_550-472-672/sperner.pdf · Proof of Sperner’s Lemma We create a graph out of the triangles. I \Outside"](https://reader034.fdocuments.us/reader034/viewer/2022050504/5f963b15c5d236713860cea9/html5/thumbnails/4.jpg)
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).
![Page 5: Sperner's Lemma: An Application of Graph Theory @let@token ...abasu9/AMS_550-472-672/sperner.pdf · Proof of Sperner’s Lemma We create a graph out of the triangles. I \Outside"](https://reader034.fdocuments.us/reader034/viewer/2022050504/5f963b15c5d236713860cea9/html5/thumbnails/5.jpg)
A Problem on triangles
Color points with 3colors using two rules:
![Page 6: Sperner's Lemma: An Application of Graph Theory @let@token ...abasu9/AMS_550-472-672/sperner.pdf · Proof of Sperner’s Lemma We create a graph out of the triangles. I \Outside"](https://reader034.fdocuments.us/reader034/viewer/2022050504/5f963b15c5d236713860cea9/html5/thumbnails/6.jpg)
A Problem on triangles
Color points with 3colors using two rules:
1. Corners getdifferent colors
![Page 7: Sperner's Lemma: An Application of Graph Theory @let@token ...abasu9/AMS_550-472-672/sperner.pdf · Proof of Sperner’s Lemma We create a graph out of the triangles. I \Outside"](https://reader034.fdocuments.us/reader034/viewer/2022050504/5f963b15c5d236713860cea9/html5/thumbnails/7.jpg)
A Problem on triangles
Color points with 3colors using two rules:
1. Corners getdifferent colors
2. Edge gets colors ofits endpoints
![Page 8: Sperner's Lemma: An Application of Graph Theory @let@token ...abasu9/AMS_550-472-672/sperner.pdf · Proof of Sperner’s Lemma We create a graph out of the triangles. I \Outside"](https://reader034.fdocuments.us/reader034/viewer/2022050504/5f963b15c5d236713860cea9/html5/thumbnails/8.jpg)
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.
![Page 9: Sperner's Lemma: An Application of Graph Theory @let@token ...abasu9/AMS_550-472-672/sperner.pdf · Proof of Sperner’s Lemma We create a graph out of the triangles. I \Outside"](https://reader034.fdocuments.us/reader034/viewer/2022050504/5f963b15c5d236713860cea9/html5/thumbnails/9.jpg)
Simple observation about line segments
I Start with any line segment.
![Page 10: Sperner's Lemma: An Application of Graph Theory @let@token ...abasu9/AMS_550-472-672/sperner.pdf · Proof of Sperner’s Lemma We create a graph out of the triangles. I \Outside"](https://reader034.fdocuments.us/reader034/viewer/2022050504/5f963b15c5d236713860cea9/html5/thumbnails/10.jpg)
Simple observation about line segments
I Start with any line segment.
I Mark any subset of points on the line segment which includeend points.
![Page 11: Sperner's Lemma: An Application of Graph Theory @let@token ...abasu9/AMS_550-472-672/sperner.pdf · Proof of Sperner’s Lemma We create a graph out of the triangles. I \Outside"](https://reader034.fdocuments.us/reader034/viewer/2022050504/5f963b15c5d236713860cea9/html5/thumbnails/11.jpg)
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.
![Page 12: Sperner's Lemma: An Application of Graph Theory @let@token ...abasu9/AMS_550-472-672/sperner.pdf · Proof of Sperner’s Lemma We create a graph out of the triangles. I \Outside"](https://reader034.fdocuments.us/reader034/viewer/2022050504/5f963b15c5d236713860cea9/html5/thumbnails/12.jpg)
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
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Proof of Sperner’s Lemma
We create a graph outof the triangles.
![Page 14: Sperner's Lemma: An Application of Graph Theory @let@token ...abasu9/AMS_550-472-672/sperner.pdf · Proof of Sperner’s Lemma We create a graph out of the triangles. I \Outside"](https://reader034.fdocuments.us/reader034/viewer/2022050504/5f963b15c5d236713860cea9/html5/thumbnails/14.jpg)
Proof of Sperner’s Lemma
We create a graph outof the triangles.
I Put a vertex foreach smalltriangle.
![Page 15: Sperner's Lemma: An Application of Graph Theory @let@token ...abasu9/AMS_550-472-672/sperner.pdf · Proof of Sperner’s Lemma We create a graph out of the triangles. I \Outside"](https://reader034.fdocuments.us/reader034/viewer/2022050504/5f963b15c5d236713860cea9/html5/thumbnails/15.jpg)
Proof of Sperner’s Lemma
We create a graph outof the triangles.
I Connect vertices⇔ correspondingtriangles sharemulti-colorededge.
![Page 16: Sperner's Lemma: An Application of Graph Theory @let@token ...abasu9/AMS_550-472-672/sperner.pdf · Proof of Sperner’s Lemma We create a graph out of the triangles. I \Outside"](https://reader034.fdocuments.us/reader034/viewer/2022050504/5f963b15c5d236713860cea9/html5/thumbnails/16.jpg)
Proof of Sperner’s Lemma
We create a graph outof the triangles.
I Put extra vertexfor “outside”.
![Page 17: Sperner's Lemma: An Application of Graph Theory @let@token ...abasu9/AMS_550-472-672/sperner.pdf · Proof of Sperner’s Lemma We create a graph out of the triangles. I \Outside"](https://reader034.fdocuments.us/reader034/viewer/2022050504/5f963b15c5d236713860cea9/html5/thumbnails/17.jpg)
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.
![Page 18: Sperner's Lemma: An Application of Graph Theory @let@token ...abasu9/AMS_550-472-672/sperner.pdf · Proof of Sperner’s Lemma We create a graph out of the triangles. I \Outside"](https://reader034.fdocuments.us/reader034/viewer/2022050504/5f963b15c5d236713860cea9/html5/thumbnails/18.jpg)
Proof of Sperner’s Lemma
We create a graph outof the triangles.
I “Outside” vertexhas odd degree byline segmentobservation.
![Page 19: Sperner's Lemma: An Application of Graph Theory @let@token ...abasu9/AMS_550-472-672/sperner.pdf · Proof of Sperner’s Lemma We create a graph out of the triangles. I \Outside"](https://reader034.fdocuments.us/reader034/viewer/2022050504/5f963b15c5d236713860cea9/html5/thumbnails/19.jpg)
Proof of Sperner’s Lemma
We create a graph outof the triangles.
I “Outside” vertexhas odd degree byline segmentobservation.
I No degree 1 innervertex.
![Page 20: Sperner's Lemma: An Application of Graph Theory @let@token ...abasu9/AMS_550-472-672/sperner.pdf · Proof of Sperner’s Lemma We create a graph out of the triangles. I \Outside"](https://reader034.fdocuments.us/reader034/viewer/2022050504/5f963b15c5d236713860cea9/html5/thumbnails/20.jpg)
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.
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Questions?