Math 184A Second Exam 24 November 2003
•Please put your name and ID number on your blue book.
•The exam is CLOSED BOOK, but you may have a page of notes.
•Calculators are NOT allowed.
•You must show your work to receive credit.
1. (16 pts.) An oriented graph is a simple graph in which each edge has been given
a direction. In other words, given vertices xand ywith x6=y, exactly one of the
following is true:
•There is no edge between xand y.
•There is an edge from xto y.
•There is an edge from yto x.
Obtain formulas for
(a) the number of oriented graphs with vertex set n; that is, the number of n-vertex
oriented graphs;
(b) the number of n-vertex oriented graphs having exactly kedges.
2. (a) (2 pts.) Sketch the simple graph G= (V, E ) where
V={a, b, c, d, e}E={a, b},{a, c},{a, d},{b, c},{d, e}.
(b) (6 pts.) Compute the chromatic polynomial PG(x).
(c) (2 pts.) How many ways can Gbe properly colored if 5 colors are available?
3. (24 pts.) There are an= 2nn-long sequences of zeroes and ones, including the empty
sequence, and so A(x) = Panxn= (1 −2x)−1. (You do not need to derive this.)
Let fnbe the number of such sequences that do not contain the pattern 11100.
Let F(x) = Pfnxn.
(a) Derive either of the two formulas
A(x) = F(x) + A(x)x5F(x)A(x) =
∞
X
t=0
(F(x)x5)tF(x).
(Both formulas are correct. Which you derive will depend on how you think
about the problem.)
(b) Using either of the formulas in (a) and the formula for A(x), find polynomials
P(x) and Q(x) so that F(x) = P(x)
Q(x); for example, F(x) might be 7
23−x9.
(c) Using (b) or otherwise, obtain a simple recursion for fnfor n≥5. Don’t worry
about initial conditions.
Final Exam in Center 113
END OF EXAM
