1. How many different subgraphs that are isomorphic to C6 can be found in K3,3? In
K3,4? In K4,4?
Label the vertices in the two partite sets A={1,2,3} and B={a,b,c}.
The vertex labeled 1 can be connected to three different vertices in set B, say a. Then
the vertex labeled a can be connected to the two vertices in set A-{1}, say 2. Then the
vertex labeled 2 can be connected to two vertices in B-{a}, say b. Now we have no
choices left and we have the cycle 1-a-2-b-3-c. This yields a possibility of 3(2)(2) =
12 cycles of length 6. But, each cycle is counted twice – itself and the cycle labeled in
the reverse order. Therefore, there are 12/2 = 6 different subgraphs isomorphic to C6.
 4  3 
There are    6  24 subgraphs isomorphic to C6 in K3,4.
 3  3 
 4  4 
There are    6  96 subgraphs isomorphic to C6 in K4,4.
 3  3 
2. Find the number of spanning trees of the graph below.
Find all the spanning trees that
 3  3 
a) contain the edge e but not f and g:     9
 2  2 
 3  3 
b) contain the edge f but not e and g:     9
 2  2 
 3  3 
c) contain the edge g but not f and e:     9
 2  2 
 3
d) contain the edges e and f but not g: 2   2  12
 2
 3
e) contain the edges e and g but not f: 2   2  12
 2
 3
f) contain the edges g and f but not e: 2   2  12
 2
 3   3   3  2 
g) contain the edges e, f and g:           12
 2   2   1  1 
There are a total of 75 spanning trees.
3. Redraw the graph below so that it is represented as a cycle with chords. To what
well known graph is this graph isomorphic?
Q3
4. Draw two copies of K4 and name the vertices {1,2,3,4} and {5,7,6,8}in a
clockwise manner respectively.. Connect vertices 1 to 6, 2 to 5, 3 to 8 and 4 to 7.
Find the complement of this graph. To what well known graph is this complement
isomorphic?
Q3
5. We define the degree set of a graph G as the set of the different degrees of the
vertices of G. Find a graph with 8 vertices and with the degree set {3,4,5,7}.
6. Write a statement that gives necessary and sufficient conditions so that the line
graph of a graph G has edge-chromatic number 2.
Prove your statement.
The line graph G has edge-chromatic number 2 if and only if G is an even cycle or a
path.
Let L(G) be the line graph of G. If the edge-chromatic number of L(G) is 2, then any
vertex in L(G) has maximum degree 2. The only such graphs are cycles and paths. If
L(G) is an odd cycle then it cannot have edge-chromatic number 2.
If G is an even-cycle, then L(G) is an even-cycle and has edge-chromatic number 2. If
G is a path with n edges, the L(G) is a path with n-1 edges. All paths have edgechromatic number 2.
7. Label the vertices of K7 with the numbers 1,…,7. Show a decomposition of K7 into 3cycles by naming the 3-cycles with the numbers on the vertices. Do this by arranging the
numbers in an ascending. Finally, list the 3-cycles in ascending order.
How many times does each vertex appear altogether in this list?
One such decomposition is as follows:
127
134
156
236
245
357
467
Each vertex appears three times.
8. Prove the statement:
If Kp is decomposable into 3-cycles, then p is odd and p  0 mod 3 or p  1mod 3 .
If a 3-cycle appears at a vertex of Kp, then it accounts for two edges at that vertex, as does
any other 3-cycle at the vertex. That is, the vertex has even degree and p must be odd.
p ( p  1)
The number of edges of Kp must be divisible by 3. Since |E(Kp)| =
, this quantity
2
must be divisible by 3. So, 3 divides p  p  0 mod 3 or 3 divides p-1  p  1mod 3 .
9. Draw the line graph of W4. What is the largest clique of L(W4)? That is, what is the
complete graph with the greatest number of vertices that is also a subgraph of L(W4)?
Find the complement of L(W4). What is the largest clique of the complement of L(W4)?
The largest clique of L(W4) is K4.
The largest clique of the complement of L(W4) is K2.
10. Use the turning trick to decompose K6 into three isomorphic subgraphs.
11. Prove the following statement.
If the degree of every vertex of a graph G is even, the G contains no bridge.
Proof by contradiction. Suppose G contains the bridge (u,v). If we remove the bridge, we
have two components and the degree of u and the degree of v is now odd. So, in each
component all vertices have even degree except for one vertex. But, by Theorem 1.1.1,
the sum of the degrees must be even which is not the case.
12. Find three nonisomorphic graphs with the degree sequence 5,3,2,2,1,1,1,1. (Note.
One is disconnected.)