CSCI 190 Exam III Practice
1) (6 points) Prove that a connected simple graph of order greater than one with no circuits has
at least one vertex of degree 1.
2) ( 6 points) 30 of the 50 US state flags have blue as a background color, 12 have stripes, 26
exhibit a plant or animal, 9 have both blue in the background and stripes, 23 have both blue in
the background and feature a plant or animal, and three have both stripes and a plant or
animal. . Two state flags have blue in the background, stripes, and plant or animal. How
many state flags have no blue in the background, no stripes, and no plant or animal featured?
3) (4points each) Determine if each of the following degree sequences on 6 vertices are possible.
Sketch a graph with the given degree sequence if possible. If not possible, explain briefly
why not. Justify your answer.
a) 1, 2, 3, 4, 5, 5
b) 3,3,3,2,2,2
4) ( 4 points each) Use the following adjacency matrix for the following questions
0 0 0 
1 0 1 


1 1 0 
a) Construct a graph for the following adjacency matrix
b) Find the number of paths from
v1 to v3 that uses exactly 4 edges. You must apply theorem
discussed in class. DO NOT use a brute force approach.
5) Define a relation on {1, 2,3, 4,5, 6, 7,8} as follows:
aRb iff a b .
You may assume R is a partial
ordering.
a) (4 points) Construct a Hasse diagram for the relation.
b) (2 points) Is R a linear ordering? Justify your answer.
c) (2 points) Is R a lattice? Justify your answer.
6) (4 points each )Use the graph below to answer the questions:
a) Is the graph bipartite?
 (G ) and minimal cut edge set.
c) Find  (G ) and minimal cut vertex set.
b) Find
7) Let G=(V,E) be a undirected graph, where V is the vertex set of G and E is the edge set of G.
Define a relation on V as follows: aRb iff a=b or there is a path from a to b.
a) (6 points) Show R is an equivalence relation.
b) (2 points) Describe the equivalence classes using a terminology from graph theory
8) (4 points) Construct all non-isomorphic undirected graphs of order 4.
9) (2 points each) Do not show work.
a) Name three invariants of an isomorphism of graphs.___________,____________,__________
b) For
R1  {(1, 2),(1,3),(2,3)}, R2  {(1, 2),(3,1),(2, 2)} , find R1 R2 _________________________
c) Define a relation on the set of real numbers that is both an equivalence relation and a
partial ordering._____________________________
d) (True/false) The greatest element may not be a maximal element
e) Arrange (1,100), (2,-1), (-1,23) in increasing order using Lexicographic ordering on
10) (4 points) Determine whether the following graphs are isomorphic. Justify your answer.\
11) (4 points) Find an Euler circuit if possible.
R2
12) Define a relation R in on the set of ordered pairs
R 2 as follows:
(a, b) R(c, d ) iff a 2  d 2  b 2  c 2
a) Is R reflexive?
b) Is R symmetric?
c) Is R antisymmetric?
d) Is R transitive?
e) If R is an equivalence relation, describe [(1,3)] geometrically.
13)
Let G be a simple graph with n vertices for some even number n. Show that if the minimum
degree of the vertices is greater than or equal to
n 1
, then the graph is connected.
2
14) Suppose an unlimited number of doughnuts is available in each of three different types. Use a
generating function to find the number of ways to select 10 doughnuts if we must select at
1 n   n  k  1 k
)  
least two doughnuts of each type. You may use the formula (
x
1 x
k 
k 0 
15) Show that a full binary tree having exactly k internal vertices has exactly k+1 leaves.
16) Define a relation on the set of real numbers as follows:
aRb
iff
a b  0
Determine if the relation is
a. Reflexive
b. Symmetric
c. Transitive
d. Antisymmetric
e. Irreflexive
17)
A relations on the positive real numbers is defined as follows:
R1  {(a, b) : a  b} , R2  {(a, b) : a b}
a) (2 points)Find
R1  R2
b) (3 points) describe R1
2
18) Define a relation R on the set of rational numbers as follows: )
aRb
iff
a  b is an integer.
a) (3 points) Show R is an equivalence relation.
1
2
b) (2 points) Find three elements in [ ]
19) Find all the connected components(2 points) and cut vertices(2 points) and
the graph shown below:
a.
b.
c.
.d
.f
.e
.g
20) (1 points) Is the graph shown below a tree? Justify your answer.
21) (4 points)
 (G )
(1 point) for
Find all nonisomorphic rooted trees with 4 vertices.
22) (5 points each)
a) Let R be a relation on a nonempty set A. Show that if R is transitive, then
b) Let R be a reflexive relation on a nonempty set A. Show that
R2  R
R  R2
23) Let G be a simple graph. Define a relation on the set of vertices of G as follows:
(u , v)  R iff
there is an edge connecting u and v. a) Is R reflective? b) Is R transitive? Justify your answer.
24) Is (3,3,2,1,1) a degree sequence of a simple (undirected) graph? Justify your answer.
25) Define
ak to be the number of ways to make k cents in change using nickels and quarters
only. Construct a generating function for
ak .
You must simplify the geometric series.
1 1 1


26) Consider a relation R represented by the matrix 0 1 1


0 1 1
a) Is R reflective?
b) Is R symmetric?
c) Is R antisymmetric?
d) Is R transitive? Justify your answer.
e) Can R be the adjacency matrix of a simple graph? Justify your answer.
27) Consider a Hasse diagram shown below:
a) Find all the maximal elements.
b) Find all the upper bounds of {3,6}.
c) Is this a lattice?
28) Let G=(V,E) be a simple graph. Show that if
 (G )  2 (the minimum degree of G), then G
contains a path of length at least two.
29) Consider a relation
R  {( 3,4), (4,3), (2,3), (1,1), (2,2)} on {1,2,3,4}
a) R is not symmetric. Add the minimum number of elements to R to make the relation
symmetric.
b) R is not transitive. Add the minimum number of elements to R necessary to make
the relation transitive. (use the original relation, not the relation constructed in a))
c) Is R antisymmetric? Justify your answer.
30) Define relations on the set of the natural numbers as follows:
R2  {( a, b) : a  b} , R3  {( a, b) : a  b} , R4  {( a, b) : a  b}
a) Find
R12
R1  {( a, b) : a b} ,
b) Find
R3  R4
c) Find
R2  R3
31)
R2  R
2
Show that R  R
c) Let R be a relation on a nonempty set A. Show that if R is transitive, then
d) Let R be a transitive and reflexive relation on a nonempty set A.
e) Find an example of a relation on a set such that
32) Suppose a relation R is transitive and irreflexive.
(a, b)  R 2 but (a, b) 
 R
Show it is antisymmetric