This preview shows page 1. Sign up to view the full content.
258
(b) Yes.
In
the graph at the right, the
bipartition sets are labeled
{I, 2, 3, 4} and
{A, B, C, D}.
Solutions to Exercises
1
A
:@:
C
3
1
4
6. The labelings show that the graphs are
isomorphic.
5
2
2
~\h+_~B
1
E
A
C
7. [BB] Any graph
9
with
n
vertices is a subgraph of
lCn ,
as is easily seen by joining any pair of vertices
of 9 where there is not already an edge.
8. (a) [BB] Suppose that
9 and
1i
are isomorphic graphs and
r.p: V(Q)
+
V(1i)
is the isomorphism of
vertex sets given by Definition 9.3.1. Label the edges of
9
arbitrarily,
gl,g2,
... ,gm,
and then
use
r.p
to label the edges of
as
hI, h2,
... ,
hm;
that is, if edge
gl
has end vertices
v
and
w
in
9,
let
hI
be the edge joining
r.p(v)
and
r.p(w)
in
1i.
Repeat to obtain
h2"'" hm.
Now a triangle in
9
is a set of three edges
{gi, gj, 9k}
each two of which are adjacent.
It
follows from the way we
labeled the edges of
that
{gi, gj,
gd
is a triangle in
9 if
and only if
{hi, hj , hk}
is a triangle in
. Thus, the number of triangles in each graph is the same.
This is the end of the preview. Sign up
to
access the rest of the document.
 Summer '10
 any
 Graph Theory, Sets

Click to edit the document details