UC Berkeley, Spring 2009
Solutions to Homework # 11 (due April 27)
§
9.1, #13: See diagrams.
§
9.1, #27: This is best represented by a directed multigraph which allows loops
but not duplicate edges.
§
9.2, #5: No, because
X
v
∈
V
deg
v
= 45 which is odd; but this is impossible by the
Handshaking Theorem.
§
9.2, #20: See diagrams.
§
9.2, #29: a)
K
n
has
n
vertices and
(
n
2
)
edges.
b)
C
n
has
n
vertices and
n
edges.
c)
W
n
has
N
+ 1 vertices and 2
n
edges.
d)
K
m,n
has
m
+
n
vertices and
mn
edges.
e)
Q
n
has 2
n
vertices and
n
2
n

1
edges (since each vertex has degree
n
).
§
9.2, #31: a) 3,3,3,3
b) 2,2,2,2
c) 4,3,3,3,3
d) 3,3,2,2,2
e) 3,3,3,3,3,3,3,3
§
9.2, #36: a) Not graphic; the vertex with degree 5 must be adjacent to each of
the others, but one of these has degree 0.
b) Not graphic; a simple graph with 6 vertices cannot have a vertex of
degree 6.
c) The cycle
C
6
is one such graph.
d) Not graphic: the degrees sum to an odd number, violating the Hand
shaking Theorem.
e) See diagrams.
f) See diagrams.
g) The wheel
W
5
is one such example.
1