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Qualifying Exam in Combinatorics
19 August, 2008
1.
Let Γ be a 3connected planar graph with planar dual Γ
*
. Prove or
disprove:
If
Γ
is a Cayley graph and
Γ
*
is is vertextransitive, then
Γ
*
is a Cayley
graph.
If you believe the statement to be true, give a proof, or at least
discuss what must be done to give a correct proof. If you believe the
statement to be false, then give a counterexample, proving that exactly
one of Γ and Γ
*
is a Cayley graph.
2.
(a) Starting with the deﬁnition of
derangement
, derive an explicit
formula for the
n
th derangement number
D
n
. (Assume
D
0
= 1 and
D
1
= 0.)
(b) Obtain an exponential generating function for the sequence
{
D
n
:
n
≥
0
}
.
(c) Give a combinatorial proof (not an algebaic proof) of the recurrence
D
n
= (
n

1)(
D
n

1
+
D
n

2
)
(
n
≥
2)
.
3.
Describe how to “complete” an aﬃne plane of order
n
to obtain
a projective plane of order
n
. Then explain what this has to do with
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This note was uploaded on 06/19/2011 for the course MATH 680 taught by Professor Na during the Spring '11 term at University of North Carolina School of the Arts.
 Spring '11
 NA
 Combinatorics

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