Download 2.2 Hamilton Circuits

2.2 Hamilton Circuits
Tucker, Applied Combinatorics, Section 2.2, Tamsen Hunter
Hamilton Circuits
Hamilton Paths

2.2 Hamilton Circuits
a
b
d
c
Definition of Hamilton Path: a path
that touches every vertex at most
once.
2.2 Hamilton Circuits
a
b
d
c
Definition of Hamilton Circuit: a
path that touches every vertex at
most once and returns to the
starting vertex.
2.2 Building Hamilton Circuits
Rule 1: If a vertex x has degree 2, both of the
edges incident to x must be part of a Hamilton
a
Circuit
b
c
d
e
f
g
h
i
j
k
The red lines indicate the
vertices with degree two.
2.2 Hamilton Circuit
Rule 2: No proper subcircuit, that is, a circuit not containing
all vertices, can be formed when building a Hamilton Circuit
a
b
h
c
i
d
g
f
e
2.2 Hamilton Circuit
Rule 3: Once the Hamilton Circuit is required to use two
edges at a vertex x, all other (unused) edges incident at x
can be deleted.
a
b
c
d
e
f
g
h
i
j
k
The red lines indicate
the edges that have
been removed.
2.2 Hamilton Circuits
Applying the
Rules One & Two
a
b
c
d
e
f
g
h
i
j
k
Rule One: a and g
are vertices of
degree 2, both of
the edges
connected to those
2 vertices must be
used.
Rule Two: You must
use all of the vertices to
make a Hamilton
Circuit, leaving out a
vertex would not form a
circuit.
2.2 Hamilton Circuits
Step One: We have
two choices leaving iij or ik if we choose ij
then Rule Three
applies.
Applying the
Rule Three
a
b
c
d
e
f
g
h
i
j
k
Step Two: Edges jf
and ik are not needed
in order to have a
Hamilton Circuit, so
they can be taken out.
Step Three: We now
have two choices
leaving j, jf or jk. If we
choose jk, then Rule
Three applies and we
can delete jf.
2.2 Hamilton Circuits
Theorem 1
A connected graph with n vertices, n >2, has a Hamilton circuit
if the degree of each vertex is at least n/2
a
b
e
c
d
2.2 Hamilton Circuits
Theorem 2
Let G be a connected graph with n vertices, and let the vertices
be indexed x1, x2,…, xn, so that deg(xi)  deg(xi+1). If for each
k  n/2, either deg (xk) > k or deg(xn+k)  n – k, then G has a
Hamilton circuit
2.2 Hamilton Circuits
Theorem 3
Suppose a planar graph G has a Hamilton circuit H. Let G be
drawn with any planar depiction, and let ri denote the number
of regions inside the Hamilton circuit bounded be i edges in
this depiction. Let r´i be the number of regions outside the
circuit bounded by i edges. Then the numbers ri and r´i satisfy
the equation
2.2 Hamilton Circuit
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b
d
4
m
6
e
a
q
6
6
l
p
o
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n
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4
4
g
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6
2.2 Hamilton Circuits
Equation in Math Type
2.2 Hamilton Circuit
Theorem 4
Every tournament has a Hamilton path.
A tournament is a directed graph obtained from a complete
(undirected) graph by giving a direction to each edge.
a
c
All of the tournaments for
this graph are;
a-d-c-b, d-c-b-a, c-b-d-a,
b-d-a-c, and d-b-a-c.
b
d
2.2 Hamilton Circuits
Class Work
Exercises to Work On (p. 73 #3)
Find a Hamilton Circuit or prove that one
doesn’t exist.
a
One answer is;
a-g-c-b-f-e-i-k-h-d-j-a
f
e
b
j
g
k
i
d
h
c
2.2 Hamilton Circuits
Class Work Exercises to Work On
Find a Hamilton circuit in the following graph. If one exists. If
one doesn’t then explain why.
a-f-b-g-c-h-d-e-a is
forced by Rule
One, and then
forms a subcircuit,
violating Rule Two.
a
e
d
f
b
i
h
g
c