MATH 239 — Fall 2011
Assignment 5
DUE: NOON Friday 28 October 2011
in the drop boxes opposite the Math Tutorial
Centre MC 4067 or next to the St. Jerome’s library for the St. Jerome’s section.
1. Given a graph
G
,thel
inegraph
L
(
G
)isde±nedinthefo
l
low
ingway
:
V
(
L
(
G
)) =
E
(
G
)
,
E
(
L
(
G
)) =
{{
e
1
,e
2
}
e
1
∩
e
2

=1
}
.
Prove that the line graph
L
(
K
m,n
) to the complete bipartite graph
K
m,n
is regular and
±nd the common degree to all its vertices.
SOLUTION.
Let (
A,B
)beab
ipa
r
t
i
t
iono
f
V
(
K
m,n
), with

A

=
m
and

B

=
n
.
Let
e
∈
V
(
L
(
K
m,n
)) =
E
(
K
m,n
). We have
e
=
{
a,b
}
for some
a
∈
A
and some
b
∈
B
.
Since
{
°
}∈
E
(
K
m,n
)and
{
a
°
,b
E
(
K
m,n
)fo
ra
l
l
b
°
∈
B
and
a
°
∈
A
, and since
those are all the edges containing
a
and
b
,wehave
{
e
°
∈
E
(
K
m,n
)

e
°
∩
e

}
=
°
{
°
}
b
°
∈
B
\{
b
}
±
∪
°
{
a
°
a
°
∈
A
a
}
±
.
Hence deg(
e
)=

B
−
1+

A
1=
m
+
n
−
2. Since that result is independent of
e
,
L
(
K
m,n
)is(
m
+
n
−
2)regular.
2. A sequence of decreasing integers is called
graphic
if it corresponds to the degrees of
the vertices a graph. Which of these sequences are graphic? Justify your answer.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '13
 Math, Combinatorics, Graph Theory, Vertex, line graph, vertices, Bipartite graph, Perfect graph

Click to edit the document details