math algebra questions
Exercise11.1
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 twoelement 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 (loopfree) 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.
Answer:
Exercise 11.5
7. a) For n ≥ 3, how many different Hamilton cycles are there
in the complete graph Kn?

b) How many edgedisjoint Hamilton cycles are there in
K21
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.