MATH 1050, Homework 5
Kiumars Kaveh
September 27, 2013
Due date: Wednesday October 2, 2013
Problem 1:(a) Let G = (V, E ) be a graph. For a vertex v V let d(v )
denote the degree of v , i.e. the number of edges connected to v . Show:
v V
d(v ) = 2|E |.
(b)
Chapter 2. Strings, Sets, and Binomial Coefcients
In fact, I tried it out with two integers, each having more than one thousand digits. It
found the product in about one second. Ever the skeptic, Zori said You mean you
carefully typed in two integers of t
MATH 1050, Homework 2 Solutions
Kiumars Kaveh
September 4, 2013
Problem 1: Let n and k be positive integers. Give a formula for the number of solutions of n = x1 + + xk where the xi are positive integers (not
equal to zero). (We solved a similar problem i
MATH 1050, Homework 6
Kiumars Kaveh
October 3, 2013
Due date: Wednesday October 9, 2013
Problem 1: Let v , w be two vertices of a graph G. Show that if there is
a walk between v and w then there is a path between v and w. (Hint: take
the shortest walk bet
MATH 1050, Homework 4
Kiumars Kaveh
September 18, 2013
Due date: Wednesday September 25, 2013
Problem 1: Suppose you have a set A with 10 (distinguishable) elements
e.g. A = cfw_1, . . . , 10. In how many ways you can color the elements of A
with 4 colors
MATH 1050, Homework 3
Kiumars Kaveh
October 12, 2013
Due date: Wednesday September 18, 2013
Problem 1: The binomial expansion theorem states that (a+b)n = n
i=0
(a) Extend this theorem to sum of three numbers, that is, prove that:
(a + b + c)n =
i+j +k=n;
MATH 1050, Homework 7
Kiumars Kaveh
October 18, 2013
Due date: Wednesday October 23, 2013
s
Problem 1: Prove by induction that R(r, s) r+2 . Hint: recall the
r1
inductive proof of Ramseys theorem stating that R(r, s) exists.
Problem 2:(a) (A theorem of Sc
MATH 1050, Homework 8
Kiumars Kaveh
October 25, 2013
Due date: Friday November 1st, 2013
Problem 1: In the weighted graph below, apply Kruskals algorithm to
nd a minimal spanning tree. In each step of the algorithm show what is
the forest F and the set of
Department of Mathematics
University of Pittsburgh
MATH 1050 (Combinatorics)
Midterm, Fall 2013
Instructor: Kiumars Kaveh
Last Name:
Student Number:
First Name:
TIME ALLOWED: 50 MINUTES. TOTAL: 50+2
NO AIDS ALLOWED. WRITE SOLUTIONS ON THE SPACE PROVIDED.
MATH 1050, Homework 10
Kiumars Kaveh
November 28, 2013
Due date: Wednesday December 4, 2013
Problem 1: In the graph below use Phillip Halls theorem to prove that
there is no perfect matching.
Problem 2: Draw the network associated to the bipartite graph b
MATH 1050, Homework 9
Kiumars Kaveh
November 1, 2013
Due date: Friday November 8th, 2013
Problem 1: (a) Write down the linear programming problem corresponding to the max ow problem in the network below. (b) Also write down
the linear programming problem