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
, the line graph
L
(
G
) is defined in the following way:
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
find the common degree to all its vertices.
SOLUTION.
Let (
A, B
) be a bipartition of
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
{
a, b
}
∈
E
(
K
m,n
) and
{
a
, b
}
∈
E
(
K
m,n
) for all
b
∈
B
and
a
∈
A
, and since
those are all the edges containing
a
and
b
, we have
{
e
∈
E
(
K
m,n
)
 
e
∩
e

= 1
}
=
{
a, b
} 
b
∈
B
\ {
b
}
∪
{
a
, b
} 
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.
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 Fall '13
 Math, Combinatorics, Graph Theory, Vertex, line graph, vertices, Bipartite graph, Perfect graph

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