Section 1.1 : Graph Models
Denition: A graph G = (V, E) consists of a nite set V of vertices and a set
E of edges joining dierent pairs of distinct vertices.
Denition: A path P is a sequence of distinct vertices, written P = x1 x2 xn
with each pair of con
MATH 3034, Spring 2012
Show all your work. No calculators allowed.
1. Show that the following two statements are logically equivalent:
P (Q R)
(P Q) R
Do this without lling in a truth table. Do it using the basic equivalencies you learned
Practice for Test #2
MATH 3034, Fall 2012
1. Let a, b, m1 , n1 , m2 , and n2 be integers such that
m1 a + n1 b = 60;
m2 a + n2 b = 45; and
3 divides a and b, but 3 = gcd(a, b).
Determine (with justication) the value of d = gcd(a, b).
2. Find the solution
1. For every real number r, set Ar = cfw_(x, y ) R2 : y = x2 + r.
(a) Give an elementwise proof that R2
(b) Give a contrapositive proof of the following statement:
Statement: For all real numbers r and s, if r = s then Ar As = .
1. Prove by induction on n: If n 2, then
Note: On the left-hand side is a product, not a sum. First do the base case. Then write down the k -case and say what
you need to show in the induction step. Then comple
Review 2 : Sections 1.1-1.4, 2.1-2.4 : Part 1
Example: Find a maximal independent set and a minimum edge cover for the
graph in Figure 1.2 on page 5. Prove that no larger independent set exists.
Example: Are the two graphs in 6a isomorphic? Prove your ans
Review for Test 2 Part 1
1. Given A = cfw_ , 1, 2, cfw_1, cfw_2 , B = cfw_ cfw_1, cfw_2, cfw_1, 2, and C = cfw_
Mark each of the following true or false.
, 1, 2, cfw_1, cfw_2, cfw_1, 2 ,
Test 1 Review - Class
1. The following items give denitions of terms. Tell me (in english) what the
denition is for the OPPOSITE of the dened term. For example, if the term
is transitive .what does it mean to NOT be transitive? These negations
should be U
1. (a) A set F of functions on R is NOT equicontinuous at a point x0 R provided
there exists a positive real number such that for all positive real numbers
there exists a function f F and a real number x such that |x x0 | < and
|f (x) f (x0 )| .
(b) A re
REVIEW FOR FINAL EXAM MATH 3034
1. In each of (a) i (d) give, in written form, both the negation and the contrapositive
of the given implication. (Label your answers as the negation and the contrapositive,
a For ever real number :1: if 0 <
Section 2.4 : Coloring Theorems
Theorem 1: A graph G is 2colorable if and only if all circuits have even length.
Proof: Recall that a graph G is bipartite if and only if every circuit in G has
even length. Finally, a graph G is bipartite if and only if i
Section 2.3 : Graph Coloring
Denition: A coloring of a graph G assigns colors to the vertices of G so that
adjacent vertices are given different colors.
Denition: The minimal number of colors required to color a given graph is
called the chromatic numb
Section 2.2 : Hamilton Circuits and Paths
Denition: A Hamilton circuit is a circuit that Visits each vertex in a graph
exactly once (except the start / end vertex).
Denition: A Hamilton path is a path that visits each vertex in a graph exactly
Section 1.3 : Edge Counting
Example: Prove that in any graph, the sum of the degrees of all vertices is
equal to twice the number of edges.
Example: Suppose we want to construct a graph of 20 edges and have all
vertices of degree 4. How many vertices
Review 3 : Sections 1.1-1.4, 2.1-2.4 : Summary
1. If I is an independent set, than V
2. If C is an edge cover, than V
I is an edge cover.
C is an independent set.
3. G1 and G2 are isomorphic if and only if G1 and G2 (the complements of G