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7. a) How many paths of length 5 are there in the complete

bipartite graph K3,7? (Remember that a path such

as v1 v2 v3 v4 v5 v6 is considered to be the

same as the path v6 v5 v4 v3 v2 v1.)

How many paths of length p are there in the complete bipartite

graph Km,n?

8. LetX _ {1, 2, 3, . . . , n}, where n 2. Construct the loopfree

undirected graph G _ (V , E) as follows:

(V ): Each two-element subset of X determines a vertex

of G.

(E): If v1, v2 V correspond to subsets {a, b} and {c, d},

respectively, of X, draw the edge {v1, v2} in G when

{a, b} ∩ {c, d} _ .

a) Show that G is an isolated vertex when n _ 2 and that

G is disconnected for n _ 3, 4.

b) Show that for n 5, G is connected. (In fact, for all

v1, v2 V , either {v1, v2} E or there is a path of length 2

connecting v1 and v2.)

6. Find all (loop-free) nonisomorphic undirected graphs with

four vertices. How many of these graphs are connected?

Exercise 11.2

13. Let G be a cycle on n vertices. Prove that G is selfcomplementary

if and only if n _ 5.

Exercise 11.3

21. Determine the value(s) of n for which the complete graph

Kn has an Euler circuit. For which n doesKn have an Euler trail

but not an Euler circuit?

Exercise 11.4

14. Determine which of the graphs in Fig. 11.69 are planar. If

a graph is planar, redraw it with no edges overlapping. If it is

nonplanar, find a subgraph homeomorphic to either K5 or K3,3.


Exercise 11.5

7. a) For n 3, how many different Hamilton cycles are there

in the complete graph Kn?

b) How many edge-disjoint Hamilton cycles are there in


c) Nineteen students in a nursery school play a game each

day where they hold hands to form a circle. For how many

days can they do this with no student holding hands with

the same playmate twice?

Exercise  12.1

7. Give an example of an undirected graphG _ (V , E) where

|V | _ |E| + 1 but G is not a tree.

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