Complete bipartite graphs. A complete bipartite graph is a graph with the property that the vertices can be divided into two sets A and B and each vertex in a set A is adjacent to each of the vertices in set B . There are no other edges! If there are m vertices in set A and n vertices in set B, the complete bipartite graph is written as K m , n . F i g u r e 6 - 5 2 shows a generic bipartite graph. F i g u r e 6 - 5 2 a. For n > 1 , the complete bipartite graphs of the form K n , n all have Hamilton circuits. Explain why. b. If the difference between m and n is exactly 1 ( i .e ., | m − n | = 1 ) , the complete bipartite graph K m , n has Hamilton path. Explain why. c. When the difference between m and n is more than 1, then the complete bipartite graph K m , n has neither a Hamilton circuit nor a Hamilton path. Explain why.
Complete bipartite graphs. A complete bipartite graph is a graph with the property that the vertices can be divided into two sets A and B and each vertex in a set A is adjacent to each of the vertices in set B . There are no other edges! If there are m vertices in set A and n vertices in set B, the complete bipartite graph is written as K m , n . F i g u r e 6 - 5 2 shows a generic bipartite graph. F i g u r e 6 - 5 2 a. For n > 1 , the complete bipartite graphs of the form K n , n all have Hamilton circuits. Explain why. b. If the difference between m and n is exactly 1 ( i .e ., | m − n | = 1 ) , the complete bipartite graph K m , n has Hamilton path. Explain why. c. When the difference between m and n is more than 1, then the complete bipartite graph K m , n has neither a Hamilton circuit nor a Hamilton path. Explain why.
Complete bipartite graphs. A complete bipartite graph is a graph with the property that the vertices can be divided into two sets A and B and each vertex in a set
A
is adjacent to each of the vertices in set
B
. There are no other edges! If there are m vertices in set A and n vertices in set B, the complete bipartite graph is written as
K
m
,
n
.
F
i
g
u
r
e
6
-
5
2
shows a generic bipartite graph.
F
i
g
u
r
e
6
-
5
2
a. For
n
>
1
, the complete bipartite graphs of the form
K
n
,
n
all have Hamilton circuits. Explain why.
b. If the difference between m and n is exactly 1
(
i
.e
.,
|
m
−
n
|
=
1
)
,
the complete bipartite graph
K
m
,
n
has Hamilton path. Explain why.
c. When the difference between m and n is more than 1, then the complete bipartite graph
K
m
,
n
has neither a Hamilton circuit nor a Hamilton path. Explain why.
By considering appropriate series expansions,
ex · ex²/2 . ¸²³/³ . . ..
=
= 1 + x + x² +……
when |x| < 1.
By expanding each individual exponential term on the left-hand side
and multiplying out, show that the coefficient of x 19 has the form
1/19!+1/19+r/s,
where 19 does not divide s.
Let
1
1
r
1+
+ +
2 3
+
=
823
823s
Without calculating the left-hand side, prove that r = s (mod 823³).
For each real-valued nonprincipal character X mod 16, verify that
L(1,x) 0.
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