*
David Craft
Mathematics
•
Combinatorics
•
Graph Theory
•
Topological Graph Theory
A graph is a set of vertices (or points)
A graph is a set of vertices (or points)
together with a set of vertex-pairs called edges.
A graph is a set of vertices (or points)
together with a set of vertex-pairs called edges.
Graph Theory is the study of graphs.
An imbedding or embedding (or proper drawing)
of a graph is one in which edges do not cross.
An imbedding or embedding (or proper drawing)
of a graph is one in which edges do not cross.
NOT an imbedding
An imbedding or embedding (or proper drawing)
of a graph is one in which edges do not cross.
NOT an imbedding
An imbedding
An imbedding or embedding (or proper drawing)
of a graph is one in which edges do not cross.
NOT an imbedding
An imbedding
Topological Graph Theory is the study of imbeddings
of graphs in various surfaces or spaces
Orientable surfaces
sphere
S0
(without boundary):
Orientable surfaces
sphere
S0
(without boundary):
torus
S1
Orientable surfaces
sphere
S0
(without boundary):
torus
S1
2-torus
S2
Orientable surfaces
sphere
S0
(without boundary):
torus
S1
n-torus
Sn
2-torus
S2
Orientable surfaces
sphere
S0
(without boundary):
torus
S1
2-torus
S2
The surface Sn
is said to have
genus n
n-torus
Sn
Some graphs cannot be imbedded in the sphere…
?
?
?
Some graphs cannot be imbedded in the sphere…
?
?
…but all can be imbedded in
in a surface of high enough
genus.
The main problem in topological graph theory:
Given a graph G, determine the smallest
genus n so that G imbeds in Sn.
The main problem in topological graph theory:
Given a graph G, determine the smallest
genus n so that G imbeds in Sn.
For G =
the answer is n = 1.
The main problem in topological graph theory:
Given a graph G, determine the smallest
genus n so that G imbeds in Sn.
For G =
For G =
the answer is n = 1.
the answer is n = 3.