Sperner’s Lemma: An Application of Graph
Theory
AMS 550.472/672: Graph Theory
Spring 2016
Johns Hopkins University
A Problem on triangles
I
Take any plane
triangle
A Problem on triangles
I
Take any plane
triangle.
I
Mark any finite
subset of points,
including corners.
A Problem on triangles
I
Take any plane
triangle.
I
Mark any finite
subset of points,
including corners.
I
Break up into
smaller triangles
(any way you like).
A Problem on triangles
Color points with 3
colors using two rules:
A Problem on triangles
Color points with 3
colors using two rules:
1. Corners get
different colors
A Problem on triangles
Color points with 3
colors using two rules:
1. Corners get
different colors
2. Edge gets colors of
its endpoints
A Problem on triangles
Color points with 3
colors using two rules:
1. Corners get
different colors
2. Edge gets colors of
its 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 include
end points.
Simple observation about line segments
I
Start with any line segment.
I
Mark any subset of points on the line segment which include
end points.
I
Color points using two colors such that end points get
different 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 out
of the triangles.
Proof of Sperner’s Lemma
We create a graph out
of the triangles.
I
Put a vertex for
each small
triangle.
Proof of Sperner’s Lemma
We create a graph out
of the triangles.
I
Connect vertices
⇔ corresponding
triangles share
multi-colored
edge.
Proof of Sperner’s Lemma
We create a graph out
of the triangles.
I
Put extra vertex
for “outside”.
Proof of Sperner’s Lemma
We create a graph out
of the triangles.
I
Put edges between
“outside” vertex
and inner vertex if
inner triangle has
multi-colored
boundary edge.
Proof of Sperner’s Lemma
We create a graph out
of the triangles.
I
“Outside” vertex
has odd degree by
line segment
observation.
Proof of Sperner’s Lemma
We create a graph out
of the triangles.
I
“Outside” vertex
has odd degree by
line segment
observation.
I
No degree 1 inner
vertex.
Proof of Sperner’s Lemma
We create a graph out
of the triangles.
I
“Outside” vertex
has odd degree by
line segment
observation.
I
No degree 1 inner
vertex.
By degree-sum formula, there are an odd number (therefore, at
least 1) of degree 3 inner vertices = “completely” colored
triangles.
Questions?