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.
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 Summer '10
 any
 Graph Theory, Sets, Vertex, Trigraph, vertices, isomorphic graphs, 1i

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