CS6702 – GRAPH THEORY AND APPLICATIONS

CS6702 GRAPH THEORY AND APPLICATIONS L T P C

3 0 0 3

OBJECTIVES:

The student should be made to:

Be familiar with the most fundamental Graph Theory topics and results. Be exposed to the techniques of proofs and analysis.

UNIT I INTRODUCTION 9

Graphs – Introduction – Isomorphism – Sub graphs – Walks, Paths, Circuits –Connectedness –

Components – Euler graphs – Hamiltonian paths and circuits – Trees – Properties of trees – Distance and

centers in tree – Rooted and binary trees.

UNIT II TREES, CONNECTIVITY & PLANARITY 9

Spanning trees – Fundamental circuits – Spanning trees in a weighted graph – cut sets – Properties of cut

set – All cut sets – Fundamental circuits and cut sets – Connectivity and separability – Network flows – 1-

Isomorphism – 2-Isomorphism – Combinational and geometric graphs – Planer graphs – Different

representation of a planer graph.

UNIT III MATRICES, COLOURING AND DIRECTED GRAPH 8

Chromatic number – Chromatic partitioning – Chromatic polynomial – Matching – Covering – Four color

problem – Directed graphs – Types of directed graphs – Digraphs and binary relations – Directed paths

and connectedness – Euler graphs.

UNIT IV PERMUTATIONS & COMBINATIONS 9

Fundamental principles of counting - Permutations and combinations - Binomial theorem - combinations

with repetition - Combinatorial numbers - Principle of inclusion and exclusion -Derangements -

Arrangements with forbidden positions.

UNIT V GENERATING FUNCTIONS 10

Generating functions - Partitions of integers - Exponential generating function – Summation operator -

Recurrence relations - First order and second order – Non-homogeneous recurrence relations - Method of

generating functions.

TOTAL: 45 PERIODS OUTCOMES:

Upon Completion of the course, the students should be able to:

Write precise and accurate mathematical definitions of objects in graph theory.

Use mathematical definitions to identify and construct examples and to distinguish examples from non-examples.

Validate and critically assess a mathematical proof.

Use a combination of theoretical knowledge and independent mathematical thinking in creative investigation of questions in graph theory.

Reason from definitions to construct mathematical proofs.

TEXT BOOKS:

1. Narsingh Deo, “Graph Theory: With Application to Engineering and Computer Science”, Prentice Hall of India, 2003.

2. Grimaldi R.P. “Discrete and Combinatorial Mathematics: An Applied Introduction”, Addison Wesley, 1994.

REFERENCES:

1. Clark J. and Holton D.A, “A First Look at Graph Theory”, Allied Publishers, 1995. 2. Mott J.L., Kandel A. and Baker T.P. “Discrete Mathematics for Computer Scientists and

Mathematicians”, Prentice Hall of India, 1996.

3. Liu C.L., “Elements of Discrete Mathematics”, Mc Graw Hill, 1985. 4. Rosen K.H., “Discrete Mathematics and Its Applications”, Mc Graw Hill, 2007.

COURSE OUTCOME

C402.1 Able to explain graph isomorphism and connectivity and describe the difference between

Eulerian and Hamiltonian graphs

C402.2 Able to analyze spanning trees, connectivity and network flow problems and understand

the concept of planar graph

C402.3 Able to identify chromatic number and understand the concept of colorings and theory

C402.4 Able to understand the problems of permutation and combination and to describe the

concepts of arrangements with forbidden

C402.5 Able to solve recurrence relations using generating functions

CS6702 - GRAPH THEORY AND APPLICATIONS

UNIT – I INTRODUCTION

1. Show that the sum of the degrees of all vertices in G is twice the number of edges in G. (U)

Proof:

Let G be a graph. Each edge of G is incident with two vertices and hence contributors 2

to the sum of degree of all the vertices of the graph G. Therefore the sum of degrees of the

vertices of a graph G is twice number of edges in G.

1

deg( ) 2n

i

i

v E

.

2. Prove that the sum of the degrees of the vertices of any finite graph is even. (U)

Proof:

Each edge ends at two vertices. If we begin with just the vertices and no edges, every

vertex has degree zero, so the sum of those degrees is zero, an even number. Now add edges

one at a time, each of which connects one vertex to another, or connects a vertex to itself (if

you allow that). Either the degree of two vertices is increased by one (for a total of two) or

one vertex's degree is increased by two. In either case, the sum of the degrees is increased by

two, so the sum remains even.

3. Explain subgraphs with suitable example. (R)

Ans: A graph g is said to be subgraph of the graph G if all the vertices and all the edges

of g are in G, and each edge of g has tha same end vertex in g as in G.

4. Define walk, path and circuit in a graph. (Nov./Dec. 2016) (R) Walks: Let ei= uiui+1 ∈G be edges of G for i ∈[1, k]. The sequence W = e1e2 . . . ek is a walk

of length k from u1to uk+1. Here ei and ei+1 are compatible in the sense that ei is adjacent to ei+1

for all i ∈[1, k −1].

Paths: Let W= e1e2 . . . ek and (ei= uiui+1) be a walk. W is a path, if ui6= uj for all i 6= j.

Circuits: W is a cycle, if it is closed, and ui6= ujfor i 6= j except that u1 = uk+1.

5. Define complement of a simple graph. (R)

Ans: The complement or inverse of a graph G is a graph H on the same vertices such

that two distinct vertices of H are adjacent if and only if they are not adjacent in G.

Ex:

https://en.wikipedia.org/wiki/If_and_only_if

6. Draw a complete bipartite graph of 3,2K and 3,4K . (U)

Ans.

7. Define a regular graph. Can a complete graph be a regular graph? (R)

Defn:

A simple graph is said to be regular if the graph contains all the vertices are of same degree.

Yes, every complete graph is a regular graph.

8. Draw the complete graph K5. (U)

Ans:

9. Prove that a complete graph with n vertices contains n(n -1) /2 edges. (U)

Proof:

This is easy to prove by induction. If n = 1, zero edges are required, and 1(1 – 0)/2 = 0. Assume

that a complete graph with k vertices has k(k -1)/2. When we add the (k + 1)st

vertex, we need to

connect it to the k original vertices, requiring k additional edges. We will then have k(k -1) /2 + k

= (k + 1)((k + 1) -1)=2 vertices, and we are done.

10. Give an example for which a graph is neither Eulerian nor Hamiltonian. (U)

Example:

11. How many vertices will the graph contain 16 edges and all vertices of degree 4? (U)

Ans: We know that

1

deg( ) 2

4 2 16

8

n

i

i

v E

Given n

n

12. Find the number of connected simple graph with four vertices. (U)

Ans: There are exactly six simple connected graphs with only four vertices.

13. Define strongly connected graph. (U)

Defn:

A digraph is said to be a strongly connected graph if and only if every pair of vertices of it is

reachable from one another.

14. Give the comparison between Euler and Hamiltonian graph. (R)

Defn:

A closed walk in a graph G containing all the edges of G is called an Euler line in G. Agraph

containing an Euler line is called an Euler graph. A cycle passing through all the vertices of a

graph is called a Hamiltonian cycle. A graph containing a Hamiltonian cycle is called a

Hamiltonian graph.

15. Define isomorphism. (U)

Ans: Two graphs G1 = (V1, E1) and G2 = = (V2, E2) are said to be isomorphic if there exists a

bijective map 1 2:f V V such that 1( , )u v E iff 2( ), ( )f u f v E .

16. Define Isomorphism and isomorphic graphs. (U)

Defn:

Consider two graphs G1 and G2 which have same number of vertices and edges.

A function f from V(G1) to V(G2) is called an isomorphism if it preserves the adjacency

between vertices. That is uv is an edge inV(G1) if and only if f(u)f(v) is an edge in V(G2)

Ex: The following graphs are isomorphic. Indeed, the required isomorphism is given

by v1→ 1, v2 → 3,v3→ 4, v4 → 2, v5 → 5.

17. State whether the following graphs are isomorphic or not. (R)

Ans: Yes the graphs are isomorphic.

18. When is a vertex of a graph said to be cut-vertex? Find all the cut-vertices of the graph G

given (U)

G :

Ans:

A vertex u is called cut vertex if deletion of the vertex makes the graph disconnected that

is G-u is disconnected. In the above graph the vertices a and e are cut vertices.

19. Define distance in a graph. (U)

Defn: The distance between two vertices in a graph is the number of edges in a shortest path

(also called a graph geodesic) connecting them.

(or) In a connected graph G, the distance d(vj,vj) between the vertices vj and vj is the

length of the shortest path between them.

Centre: The vertex v which is located more centrally than the other vertices is called as

centre.

20. Define a tree. (U)

Defn: A tree is connected graph with without any circuits or cycles.

21. Write the properties of a tree.

Ans:

A tree with n vertices has n-1 edges

There is only one path between every pair of vertices in a tree T

d

e

f

a

c b

https://en.wikipedia.org/wiki/Vertex_%28graph_theory%29https://en.wikipedia.org/wiki/Graph_%28discrete_mathematics%29https://en.wikipedia.org/wiki/Shortest_path_problem

A graph is a tree if and only if it is minimally connected.

22. Define binary tree. (U)

Ans:

A tree, in which there is exactly one vertex of degree two, and each of the remaining

vertices of degree one or three, is called a binary tree.

23. Show that the number of vertices in a binary tree is odd. (R)

Ans:

Let T be a binary tree with n vertices. T contains exactly one vertex of degree two and the

remaining vertices of degree one or three. Therefore the number of odd vertices in T is n – 1.

But the number of odd degree vertices in a graph is always even. Therefore n -1 is even.

Hence n is odd.

24. Let T be a binary tree on n vertices. Show that the number of pendent vertices in T is1

2

n . (R)

Proof:

Let p denote the number of pendent vertices in T. The number if edge in T is n – 1.

Therefore the degree sum in T = 2 (n – 1)

(p x 1) + (n – p – 1) + 2 = 2 (n – 1)

p + 3n – 3p – 3 + 2 = 2n – 2

n + 1 = 2p p = 1

2

n .

25. Define and give an example for each of the following: (R)

(i) Rooted tree (ii) Complete binary tree

Defn: Rooted tree: A rooted tree is a tree with a designated vertex called the root of the tree.

A binary for which the level order indices of the vertices from a complete interval

1, 2, 3, …..n of the integers is called a complete binary tree.

26. Define Binary Rooted tree and Balanced tree. (U)

Defn:

Binary Rooted tree: A rooted binary tree is a binary tree in which only the root is allowed

to have degree 2. The remaining nodes have degree equal to either 1 or 3.

Balanced tree: A binary tree is balanced if for each node it holds that the number of

inner nodes in the left subtree and the number of inner nodes in the right subtree differ

by at most 1. A binary tree is balanced if for any two leaves the difference of the depth

is at most 1.

27. Give an example of an Euler graph which is arbitrarily traceable. (R)

Ans:

28. What is meant by eccentricity? (Nov./Dec. 2016) (U)

Ans: A vertex v in a connected graph G has eccentricity e if the maximum of the

lengths of the shortest paths to the other vertices of G is e.

29. Discuss about Konigsberg problem and its solution. (U)

Konigsberg problem: Is it possible to design a closed walk passing through all the 7 bridges

exactly once? Or equivalently, does the graph on the right have an Eulerian walk.

Solution: Solution of this problem does not exist and this was provided by Konigsberg in

1936. Graphical representation:

30. Define spanning subgraph. (U)

Defn: A spanning subgraph for G is a subgraph of G which contains every vertex of G.

31. Determine the number of vertices for a graph G, which has 15 edges and each vertex has

degree 6. Is the graph G be a simple graph? (Nov./Dec. 2017) (U)

Solution:

We know that 1

( ) 2n

i

i

d v e

Given the number of edges 15e .

1

6 2 5n

i

e n

G cannot be a simple graph.

https://proofwiki.org/wiki/Definition:Subgraphhttps://proofwiki.org/wiki/Definition:Vertex_%28Graph_Theory%29

32. Suppose G is finite cycle-free connected graph with at least one edge. Show that G has at

least two vertices of degree 1. (Nov./Dec. 2017) (U)

Solution:

Let G be a tree with n vertices.

If n = 2 then both the vertices are of degree 1.

If n > 2, G has (n-1) edges. Then deg( ) 2( 1)v n . If we distribute these 2( 1)n degrees among n vertices, no vertex is of degree 0.

Therefore there are at least 2 vertices of degree 1.

33. For which of the following does there exist a simple graph G= (V, E) satisfying the specified

conditions? (Nov./Dec. 2018) (U)

(a) It has 3 components 20 vertices and 16 edges

(b) It is connected and has 10 edges 5 vertices and fewer than 6 cycles

(c) It has 7 vertices, 10 edges and more than two components.

Solution:

It has 7 vertices, 10 edges and more than two components.

34. The maximum degree of any vertex in a simple graph with n vertices is n-1. Give reasons.

(Nov./Dec. 2018) (U)

Solution:

In a simple graph, a vertex can form an edge with all other vertices except by itself. So the

degree of any vertex in a simple graph with n vertices is n-1.

Part – B

1. Prove that a tree with n vertices has exactly n-1 edges. (U) 2. Prove that any non-trivial tree has atleast one vertex of degree one. (U)

3. Prove that every non-trivial tree has atleast two vertices of degree 1. (U) 4. If a tree T has four vertices of degree 2, one vertex of degree 3, two vertices of degree 4 and

one vertex of degree 5, find the number of leaves in T. (R) 5. Prove that in every tree T = (V, E), │E│= │V│- 1.

(or) Prove that a graph G is a tree iff G has no cycles and │E│= │V│- 1. (U) 6. Find all the non-isomorphic spanning trees of the graph. (R) 7. If u and v are two vertices of a tree, show that there is a unique path connecting them. (U)

8. Prove that a finite graph is bipartite if and only if it contains no cycles of odd length. (U)

9. Draw the complete graph 5K with vertices EDCBA ,,,, . Draw all complete sub graph

of 5K with 4 vertices. (R)

10. Prove the following. (R)

(a) If a graph has exactly two vertices of odd degree, there must be a path joining these

two vertices.

(b) A connected graph is an Euler graph if and only if every vertex has even degree.

11. A connected graph is an Euler graph if and only if it can be decomposed into circuits. Prove

the following. (U)

a. Every tree has either one or two centers.

b. A graph is a tree if and only if it is minimally connected.

c. Number of vertices in a binary tree is always odd.

d. Number of pendent vertices in a binary tree is (n + l) / 2.

12. Prove that a graph is non-planar if and only if it contains a subgraph homeomorphic to K5 or

K3,3. (U)

13. If the distance d(x,y) between two vertices x and y in a graph is defined to be the length of

the shortest path connecting them, then prove that the distance function is a metric. (R)

14. In a complete graph having odd number of vertices, how many edge disjoint Hamiltonian

circuits exist? Prove. (R)

15. If all the vertices of an undirected graph are each of odd degree K , show that the number of

edges of the graph is a multiple of K . (U)

16. Draw the graph with 5 vertices EDCBA ,,,, such that 3deg A , B is an odd

vertex, 2deg C and D and E are adjacent. (A)

17. Find the all connected sub graph obtained from the graph given in the following figure, by

deleting each vertex. List out the simple paths from A to F in each sub graph. (R)

18. Let G be a simple graph with minimum degree of atleast two. Prove that there exists a cycle

in G . (R)

19. When is a graph said to be self-complementary? Prove that if G is self-complementary then

it has n4 or 14 n vertices. (U)

20. Define complement of a graph. Find the complement G of the following graph G . Is it true

that G is isomorphic to G (U)

G

Justify your answer?

21. Define a complete graph nK . Draw a complete graph 6K . What is the degree of each vertex

in nK ? What is the total number of edges in nK ? (R)

22. Prove that in any graph G , the number of vertices of odd degree is even. (U)

23. Find the complements of the graphs nK ; nmK , and nC .

24. Show that if G is a self-complementary simple graph with VGV vertices, then

4mod10 orV . (U)

25. If G is a simple graph with minimum degree 2G , show that G contains a cycle of

length 1 G . (U)

26. Find the maximum number of edges in a simple connected graph with n vertices. (A)

27. Draw all sub graphs of 3K containing atleast one vertex. (A)

28. If EVG , is an undirected graph with e edges, prove that i

i ev 2deg . (U)

29. Determine which of the following graphs are bipartite and which are not. If a graph is

bipartite, state if it is completely bipartite. (A)

30. Define bipartite graph. Show that if G is bipartite simple graph with p vertices and q edges

then

4

2pq . (U)

31. Let G be a simple undirected graph with adjacency matrix A with respect to the ordering

nvvvv ,.........,,, 321 . Show that number of different paths of length r from iv to jv , where

r is a positive integer, equal to the thji , entry of rA . (U)

32. Draw the graph G whose incidence matrix is given below

1110000

0011100

0000110

0101011

1000001

. (A)

33. Draw the graph whose adjacency matrix is given below

0111

1011

0100

1100

. (A)

34. The adjacency matrices of two pairs of graphs are as follows, draw those graphs 1G and 2G .

011

100

100

1GA ;

001

001

110

2GA . (A)

35. Find the adjacency matrix of the following graph G. (A)

36. Find 2A and 3A . What are your observations regarding the entries in 2A ? (A) 37. The adjacency matrices of two pairs of graph as given below. Examine the isomorphism of

G and H by finding a permutation matrix

011

100

100

GA ;

001

001

110

HA . (A)

38. Examine whether the following pair of graphs are isomorphic. If not isomorphic, give the reasons (A)

39. Using circuits, examine whether the following pairs of graphs 1G , 2G given below are

isomorphic or not? (U)

40. Determine whether the following graphs G and H are isomorphic. If the graphs are not isomorphic explain an invariant that the graphs do not share (A)

G : H :

41. Show that the following two graphs are isomorphic. (U)

G : H :

42. Establish an isomorphism between the graphs 1G and 2G given below (A)

1G : 2G :

43. Test whether the following graphs are isomorphic (A)

:G H :

44. Determine whether the graphs G and H are isomorphic (R)

G : :H

45. Examine whether the following pairs of graphs are isomorphic. Justify your answer. (U)

46. Are the two graphs given below isomorphic? Why? (A)

Fig (i): Fig (ii):

47. Define graph isomorphism and give an example of isomorphic and non-isomorphic graphs. (U)

48. Draw two isomorphic graphs having five vertices. (R)

49. Prove that the maximum number of edges in a simple disconnected graph G with n vertices

and k components is

2

1 knkn. (U)

50. Show that the maximum number of edges in a simple graph with n vertices is 2

)1( nn. (U)

(Nov./Dec. 2016)

51. Prove that a graph G is connected if and only if for any partition of V into subsets 1V and 2V

, there exists an edge joining a vertex of 1V to a vertex of 2V . (U)

52. Show that in a simple digraph every node of the digraph lies in exactly one strong component. (U)

53. If G is a simple graph with n vertices with minimum degree

2

nG , show that G is

connected.(U)

54. Let G be a simple indirected graph with n vertices. Let u and v be two non adjacent vertices

in G such that nvu degdeg in G. Show that G is Hamiltonian if and only if vuG is Hamiltonian. (U)

55. Find an Euler path or an Euler circuit, if it exists in each of the three graphs below. If it does not exist, explain why? (A)

1G 2G 3G

56. Prove that a given connected graph is Eulerian if and only if all the vertices of G are of even degree. (U)

57. Prove that the following statements are equivalent for a simple connected graph (1) G is

Eulerian (2) Every vertex of G has even degree (3) the set of edges of G can be partitioned into cycles. (U)

58. Show that nK has a Hamiltonian cycle, for 3n . What is the maximum number of edge-

disjoint Hamilton cycles possible in nK . Obtain all the edge-disjoint Hamilton cycles in 7K .

(U) 59. Define Eulerian graph and Hamiltonian graph. Give an example of a graph which is Eulerian

but not Hamiltonian and vice-versa. (R)

60. Show that the complete bipartite graph nnK , has a Hamiltonian cycle. When nnK , has

Eulerian circuit? Justify your answer. (U)

61. Define the degree of a vertex and prove that the number of vertices of odd degree is always even. (U)

62. Define an Euler path and show that if a graph G has more than two vertices of odd degrees,

then there can be no Euler path in G . (R)

63. If G is a connected graph with n vertices 3n and if the degree of each vertex is atleast

2

n, then show that G is Hamiltonian. (U)

64. Prove that if a graph G has not more than two vertices of odd degree, then there can be

Euler path in G . (U) 65. Prove that if a graph has exactly two vertices of odd degree, there must be path joining these

two vertices. (U) (Nov./Dec. 2016)

66. Prove that any two simple connected graphs with n vertices, all of degree two, are isomorphic. (U) (Nov./Dec. 2016)

67. Show that 7K has Hamiltonian graph. How many edge disjoint Hamiltonian cycles are there

in 7K ? List all the edge disjoint Hamiltonian cycles. Is it Eulerian graph? (U)

68. Give an example of a graph which contains (1) An Eulerian circuit, but not a Hamiltonian circuit. (2) An Eulerian circuit that is also a Hamiltonian circuit. (R)

69. Draw a graph which contains an Eulerian circuit that is also a Hamiltonian circuit. (A) 70. Give an example of a graph which contains (R)

(i) an Eulerian circuit that is also a Hamiltonian circuit (ii) an Eulerian circuit and a Hamiltonian circuit that are distinct (iii) an Eulerian circuit but not a Hamiltonian circuit (iv) not an Eulerian circuit but a Hamiltonian circuit (v) neither Eulerian circuit nor a Hamiltonian circuit

71. Mention some properties of trees. (U) (Nov./Dec. 2016)

72. Prove that in any tree, there are atleast two pendant vertices. (U)(Nov./Dec. 2016)

73. Show that a Hamiltonian path is a spanning tree. (U) (Nov./Dec. 2016)

74. The Figure represents a floor plan with doors between the rooms and the outside indicated.

The real estate agent would like to tour the house, starting and ending outside, by going

through each door exactly once. What is the fewest number of doors that should be added,

and where should be they be placed in order to make this tour possible? Give reasons for

your answer. (Nov./Dec. 2017) (R)

75. Define closed walk, open walk, path, and circuit. Take a graph of your choice and give an

example to each one. (Nov./Dec. 2017)

76. Nine members of committee have their dinner in round table. If no member sits near to the

same neighbour more than once, how many days can this arrangement possible? Write all

possible arrangements. (Nov./Dec. 2017)

77. State four properties of a tree graph and prove them. (Nov./Dec. 2017)

78. Draw a graph isomorphic to the graph G shown in the following figure such that no edge is

crossing each other. (Nov./Dec. 2018) (A)

79. Define walk, circuit, path and subgraph. From the graph shown in figure, draw a walk of any

length, a path of length 5, a circuit of length 4 and subgraph of 4 vertices and 5 edges. N/D

2018 (A)

e2 e7 e10

e1 e5 e4 e9

e3 e6 e8 e11

80. Seven children in a street play a game in circular arrangement. If no child hands with the

same playmate twice, how many times can this arrangement possible? Write all possible

arrangements. (Nov./Dec. 2018) (A)

81. Prove that there are at least two dependent vertices in a tree with two or more vertices. Also

prove that every tree has one or two centers. (Nov./Dec. 2018) (A)

UNIT II - TREES, CONNECTIVITY & PLANARITY

1. Define a cutvertex. (R)

Defn:

A

B

C F

D

H

E e1

Let G be a connected graph. If v is a vertex of G such that G – v is not connected then the

vertex v is called a cut vertex.

2. Define a cutedge. (R)

Defn:

Let G be a connected graph. If e is an edge of G such that G – e is not connected then the

edge e is called a cut edge.

3. Define a cutset. (R)

Defn:

Let G be a connected graph. A cut set in G is a set of edges whose removal from G leaves

the graph G disconnected provided no proper subset of these edges disconnects the graph.

4. Define edge connectivity. (R)

Defn:

Let G be a connected graph. The edge connectivity of G is the minimum of lined (edges)

whose removal results in a disconnected or a trivial graph. It is denoted by λ(G).

5. Show that the edge connectivity of a graph G cannot exceed the degree of the vertex with the

smallest degree in G. (U)

Defn.

Let vertex vibe the vertex with the smallest degree in G. Let d(vi) be the degree of vi.

Vertex vi can be separated from G by removing the d(vi) edges incident on vertex vi. Hence

the theorem.

6. Define separable graph. (R)

Defn.

A connected graph is said to be separable if its vertex connectivity is one. All other

connected graphs are called nonseparable.

7. Define vertex connectivity. (R)

Defn:

Let G be a connected graph. The minimum number of vertices whose removal results in a

disconnected or a trivial graph is called the vertex connectivity. It is denoted by k(G).

8. What is spanning tree explain with example. (U) Ans:

A tree T is said to be spanning tree of a connected graph G if T is a subgraph of G and T

contains all vertices of G.

9. If f is a face of a plane graph G and H is a subgraph of G whose edges are the boundary of f

and vertices are the end vertices of these edges, prove that f is also a face of H. (U)

Ans.

f f H and f is an open connected set in R2. If g is a connected set in R2\H with

f g and f is an open connected set in R2then f ∩ g = f, hence f is open in g. As g is

connected, f = g. So f is a face of H.

10. Define block or component. (R)

Defn.

A separable graph consists of two or more non-separable subgraphs. Each of the largest

nonseparable subgraphs is called block or component.

11. What is minimal spanning tree? (shortest distance spanning tree) (R)

Ans:

A spanning tree with the smallest weight in the weighted graph is called as minimal

spanning tree.

12. What is connectivity and separability? (R)

Ans:

A graph G is connected, if dG(u, v) < ¥ for all u, v ∈ G; otherwise, it is disconnected. The

maximal connected subgraphs of G are its connected components. Denote c(G) = the number

of connected components of G . If c(G) = 1, then G is connected.

A connected graph is said to be seperable if its vertex connectivity is one.

13. Define 1-isomorphism and 2-isomorphic. (R) (Nov./Dec. 2016)

Defn.

Two graphs G1 and G2 are said to be 1-isomorphic if they become isomorphic to each

other under repeated application that split a cutvertex to produce two disjoint sugraphs.

Defn.

Two graphs G1 and G2 are said to be 2-isomorphic if there is a one-to-one corresponding

between the set of branches B1of G1 and that B2 of G2 under this correspondence, every

loop in G1 corresponds to a loop in G2, and viceversa. It is evident that 2-isomorphism is

an equivalence relation.

14. Draw two graphs on six vertices that are 1-isomorphic but are not isomorphic. (U)

Solution: The graphs given below are clearly 1-isomorphic. However, G1has a vertex of

degree 5 while G2does not. Therefore, the graphs are not isomorphic.

15. Define planar graph. (R)

Ans. A graph is said to be planar if it can be drawn on a plane without crossovers.

16. What are the applications of planar graph? (R) (Nov./Dec. 2016)

17. Prove that K5 is non-planar. (U)

Proof: Number of vertices in K5 = │V│ = 5

Number of edges in K5 = │E│ = 10

If G is planar then │E│≤ 3│V│- 6

10 ≤ (3 x 5) – 6

10 ≤ 9, which is a contradiction.

Hence K5 is non-planar.

18. Define an embedding of a planar graph. (R)

Defn. A drawing of a geometric representation of a graph on any surface such that no edges

intersect is called embedding. An embedding of a planar graph G on a plane is called a

plane representation of G.

19. State Euler theorem in planar graph. (R)

Ans:

If G is connected planar graph then V│- │E│+│F│ = 2, where │V│ denotes the n umber of

vertices in G, │E│ denotes the n umber of edges in G and │F│ denotes the n umber of

faces or regions or planes in G.

20. What is circuit correspondence? (R)

Ans. Two graphs G1 and G2are said to have a circuit correspondence if there is a one-to-one

correspondence between the edges of G1 and G2 and a one-to-one correspondence

between the circuits of G1 and G2, such that a circuit in G1 formed by certain edges of G1

has a corresponding circuit in G2 formed by the corresponding edges of G2 and viceversa.

21. What is geometric graph? (R)

Defn.

A geometric graph is a graph G = (V, E) drawn in the plane with straight edges.

22. Define ring-sum of two graphs. (R)

Ans:

The ring sum of two graphs G1 and G2 is a graph with vertices V1UV2 and having edges

either in E1 or in E2 but not in both.

23. State Max flow min cut theorem and prove it through an example. (R)

Statement:

In a given transport network, the maximum value of a flow from s to t is equal to the

minimum value of the capacities of all the cuts in G that separate s from t.

24. In a tree, every vertex is a cut-vertex. Justify the claim. (Nov./Dec. 2017) (U)

Solution:

In a tree, every vertex of degree greater than 1 is a cut vertex. If there is no cut vertex in a

graph, then the graph has a cycle and it is not a tree.

25. A simple planar graph to which no edge can be added without destroying its planarity (while

keeping the planar simple) is a maximal planar graph. Prove that every region in a maximal

planar graph is a triangle. (Nov./Dec. 2017) (U)

Solution:

In any embedding of a planar graph G with 3n , every region is bounded by 3 edges.

Then the boundary of every region is a triangle.

26. Calculate the maximum flow between the nodes A and F in the following graph. (Nov./Dec.

2018) (U)

6

5 5 6

1 1

6 4 1

3

27. Define homeomorphic graphs and give example graphs. (Nov./Dec. 2018) (U)

Solution:

Two graphs G and G1

are homeomorphic if there is a graph isomorphism from some

subdivision of “G” to some subdivision of “G1”.

(or)

Two graphs are said to be homeomorphic if one graph can be obtained from the other by

creation of edge in series (i.e, insertion of vertex and edges)

Part – B

1. Define a spanning tree of a graph. Find all the spanning trees of the following graph shown in the

following figures. (R)

A

B

C

D

E

F

2. Prove that every connected graph has atleast one spanning tree. (U)

3. Prove that a connected graph G is a tree if and only if adding an edge between two vertices in

G creates exactly one circuit. (U)

4. Prove that a spanning tree T of a weighted graph G is a shortest spanning tree if and only if

there exists no other spanning tree at a distance of one from T whose weight is smaller than

that of T. (U)

5. Write an algorithm for finding shortest spanning tree. (A)

6. Let v be a vertex in a connected graph G. Prove that there exists a spanning tree T in G such

that the distance of every vertex from v is the same both in G and in T. (U)

7. Prove that every cutset in a connected graph G must contain atleast one branch of every

spanning tree of G. (U)

8. Prove that every connected graph has atleast one spanning tree. (U) (Nov./Dec. 2016)

9. Prove that every circuit has an even number of edges in common with any cutset. (U)

10. Prove that two graphs are 2-isomorphic if and only if they have circuit correspondence. (U)

11. Prove that in a tree, every vertex of degree greater than one is a cutvertex. (U)

12. Prove that a connected planar graph G with n vertices and m edges has exactly m – n + 2

regions in every one of its diagrams. (U)

13. Show that every simple connected planar graph G with less than 12 vertices must have a

vertex of degree ≤ 4. (U)

14. Show that in any connected planar graph with n vertices e edges and f faces, e – n- 2 = f. (U)

15. Prove that Peterson graph is non-planar. (U)

16. Prove the graphs K5 and K3,3 are non planar. (Nov./Dec. 2016) (U)

17. Using Prim’s algorithm, find a minimal spanning tree for the weighted graph shown in the

following figure (A)

18. State Kruskal 's algorithm. Apply Kruskal ' s algorithm to find a minimal spanning tree for

the following weighted graph as shown in figure (A)

19. Find the chromatic number of the complete bipartite graph Km,n and a cycle, Cn on n

vertices, n ≥ 3. (U)

20. State Krushkal's alg6rithm. Using Krushkal's algorithm find a minimal spanning tree for the

weighted graph shown in figure (A)

21. Apply Dijkstra's algorithm the diagram shown in figure and determine the shortest distance

from vertex at each of the other vertices in the directed graph. (A)

22. Explain max-flow min-cut theorem. (A) (Nov./Dec. 2016)

23. Explain about Fundamental cut set and Fundamental circuit in a graph. (A) (Nov./Dec. 2016)

24. Show that starting from any spanning tree of a graph G, every other spanning tree of G can

be G can be obtained by successive cyclic interchanges. (Nov./Dec. 2017)

25. Prove that the ring sum of any two cut-sets in a graph is either a third cut-set or an edge

disjoint union of cut-sets. (Nov./Dec. 2017)

26. Define edge vertex connectivity and edge connectivity. Give the relation between them.

(Nov./Dec. 2017)

27. Show by drawing the graphs, that two graphs with the same rank and the same nullity need

not be 2-isomorphic. (Nov./Dec. 2017)

28. State Kuratowski’s theorem and use it in order to prove the graph in the figure is non-planar.

(Nov./Dec. 2017)

29. State minimum cut maximum flow theorem. Using it calculate the maximum flow between

the nodes D and E in the following graph. The number on a line represents the capacity.

(Nov./Dec. 2017)

30. Prove that the distance between any two spanning trees is a metric. Find two different

minimum spanning trees of a graph with V = {1, 2, 3, 4} is described by

a b c d e f

1,2 1,2 1,4 2,3 3,4 3,4

. It has weights on its edges given by

a b c d e f

3 2 1 2 4 2

.

31. Prove that an Euler graph cannot have a cut-set with an odd number of edges. (N/D 2018)

(A)

32. Construct a graph G with the following properties: Edge connectivity of G = 4, vertex

connectivity of G = 3, and degree of every vertex of G5. (Nov./Dec. 2018) (A)

33. Derive the formula for the number of regions in a planar graph. G with n vertices and e

edges. Also prove that the planar graph with triangle region can have at most (3n-6) edges.

(Nov./Dec. 2018) (A)

UNIT III MATRICES, COLOURING AND DIRECTED GRAPH

Part A

1. Define chromatic number of a graph. (R)

Defn.

The smallest number of colors needed to color a graph G is called its chromatic

number, and is often denoted χ(G).

2. What is 1-chromatic of a graph? (R)

Defn.

A graph consisting of only isolated vertices is 1-chromatic.

3. What is chromatic polynomial? (R)

Defn. The chromatic polynomial counts the number of ways a graph can be colored using

no more than a given number of colors.

4. What is chromatic partition number? (R)

Ans. The chromatic partition number ( )k G is of G is the minimum number of colors needed

in a Pk- coloring of G. If ( )k G n then G is said to be (k, n)- chromatic.

5. Find the chromatic polynomial for the graph given below. (U)

6. Define matching of a graph. (R)

Defn.

A matching of graph G is a subgraph of G such that every edge shares no vertex with any

other edge. That is, each vertex in matching M has degree one.

(or)

Given a graphG = (V,E), a matching M in G is a set of pairwise non-adjacent edges; that

is, no two edges share a common vertex.

7. Define complete matching of a graph. (R)

Defn.

A perfect matching is therefore a matching containing edges (the largest possible),

meaning perfect matchings are only possible on graphs with an even number of vertices. A

perfect matching is sometimes called a complete matching or 1-factor.

8. Define coloring of a graph. (R)

Defn.

https://en.wikipedia.org/wiki/Graph_%28discrete_mathematics%29https://en.wikipedia.org/wiki/Non-adjacent

It is a way of coloring the vertices of a graph such that no two adjacent vertices share

the same color; this is called a vertex coloring.

9. Define Four color problem. (R)

Defn.

The four-color theorem states that any map in a plane can be colored using four-colors in

such a way that regions sharing a common boundary (other than a single point) do not share

the same color. This problem is sometimes also called Guthrie's problem

10. Define directed graph. (R)

Defn.

A directed graph (or digraph) is a graph, or set of vertices connected by edges, where

the edges have a direction associated with them. In formal terms, a directed graph is an

ordered pair G = (V, A) (sometimes G = (V, E)) where, V is a set whose elements are called

vertices, nodes, or points.

11. Define strongly connected and weakly connected digraphs. (R) (Nov/Dec. 2017)

Ans:

A digraph is strongly connected digraph if there is at least one directed path from every

vertex to every other vertex. It is weakly connected if its corresponding undirected graph

is connected but G is not strongly connected.

12. Write all the types of directed graph. (R)

Ans.

Simple digraph, Asymmetric digraph, Symmetric digraph, complete digraph, balanced

digraph, complete asymmetric digraph, pseudosymmetric digraph.

13. Define edge covering. (R)

Defn.

In a graph G, a set g of edges is said to cover G if every vertex in G is incident on atleast

one edge in g. A set of edges that covers a graph G is said to be an edge covering.

14. Define isomorphic digraphs. (R)

Defn.

Isomorphic graphs are defined such that they have identical behavior in terms of graph

properties.

http://mathworld.wolfram.com/Plane.htmlhttp://mathworld.wolfram.com/GuthriesProblem.htmlhttps://en.wikipedia.org/wiki/Set_%28mathematics%29https://en.wikipedia.org/wiki/Element_%28mathematics%29

15. Define simple digraph. (R)

Defn.

A directed graph that has no self loop or parallel edges is called a simple digraph.

16. Define transitive closure of a digraph with an example. (R)

Ans:

Given a directed graph, if a vertex j is reachable from another vertex i for all vertex pairs

(i, j) in the given graph that is there is a path from vertex i to j. The reach-ability matrix is

called transitive closure of a graph.

For example, consider below graph

Transitive closure of above graphs is 1 1 1 1

1 1 1 1

1 1 1 1

0 0 0 1

17. Define Euler digraph. (R)

Defn.

In a digraph G, a closed directed walk which traverses every edge G exactly once is

called a directed Euler line. A digraph containing a directed Euler line is called an Euler

digraph.

18. In a directed graph, when do we say a vertex is isolated? (R)

Ans:

In a directed graph G(V,E), the isolated vertex is the vertex having degree zero that is, a

vertex that is not an endpoint of any edge

19. What is edge coloring and edge chromatic number? (R)

Ans: A function a: E(G) → K is an edge colouring of G.

http://d1gjlxt8vb0knt.cloudfront.net/wp-content/uploads/transitiveclosure.jpg

20. Define minimal dominating set and maximal independent set. (R) (Nov./Dec. 2016)

21. Find the chromatic number of a complete graph of n vertices. (R) (Nov./Dec. 2016)

22. Show that in a directed graph where every vertex has the same number of incoming as

outgoing paths there exists an Eulerian path for the graph. (U)

Proof:

Note that each time a vertex is visited, one incoming and one outgoing node is used, so the

equality of incoming and outgoing edges is preserved.

23. Prove that a graph of n vertices is a complete graph iff its chromatic polynomial is

( ) ( 1)( 2)( 3)...........( 1)nP n . (Nov./Dec. 2017) (U)

Solution:

Let be the number of colors. First vertex can be colored in ways. Second vertex can

colored in 1 ways…………. nth vertex can be colored in ( 1)n ways.

( ) ( 1)( 2)( 3)...........( 1)nP n ways.

24. What is meant by independent set and give example graphs.

A set of vertices in a graph is said to be an independent set, if vertices of no two vertices

in a set are adjacent.

If an independent set to which no other vertex can be added without destroying its

independence property.

25. When is a digraph an Euler graph? Draw an Euler digraph. A digraph containing Eulerian trial is an Euler digraph.

Part B

1. Find the chromatic polynomial for the cycle of length 4 as shown in the following figures.

Hence find the chromatic number. (A)

2. Determine the chromatic polynomial tor the graph G in figure (A)

3. Find the chromatic polynomial for the graph shown below and also find the chromatic

number for the same. (A)

4. Prove that a graph of order (n ≥ 2) consisting of a single circuit is 2 – chromatic if n is even,

and 3 – chromatic if n is odd.(R)

5. Prove that every tree with two or more vertices is 2-chromatic. (R) (Nov./Dec. 2016)

6. Prove that a graph with atleast one edge is 2-chromatic if and only if it has no circuits of odd

length. (R)

7. Find the chromatic number of the complete bipartite graph Km,n and a cycle, Cn on n vertices,

n ≥ 3. (A)

8. Find the chromatic polynomial for the cycle of length 4. Hence find its chromatic number.

(A)

9. Prove that in a bipartite graph a complete matching of V1 and V2 exists there is a positive

integer m for which the following condition is satisfied: degree of every vertex in V1m

degree of every vertex in V2. (R)

10. Prove that a covering g of a graph is minimal if and only if g contains no paths of length

three or more. (R)

11. Prove that the vertices of every planar graph can be properly colored with five colors. (R)

12. Prove that every connected simple planar graph G is 6 colorable. (R)

13. Prove that the determinant of every square matrix of A, the incident matrix of a digraph, is 1,

-1or 0. (R)

14. Define the following with respect to a graph: i) matching ii) a cut-set. Show that the graph in

figure has a complete matching from V 1 to V 2• Obtain two complete matching. (R)

15. Define chromatic number. Prove that a graph of order consisting of a single circuit is

2 - chromatic if n is even and 3 - chromatic if n is odd.

16. Find the chromatic polynomial for the cycle of length 4 as shown in figure below. Hence find

the chromatic number. (U)

17. Prove that a graph of n vertices is a complete graph iff its chromatic polynomial is

( ) ( 1)( 2).......( 1)nP n . (U) (Nov./Dec. 2016)

18. Prove that a covering g of a graph is minimal iff g contains no paths of length three or more.

(U) (Nov./Dec. 2016)

19. Explain Euler digraph. (Nov./Dec. 2016) (A)

20. Discuss about some types of digraph with suitable example. (A) (Nov./Dec. 2016)

21. Obtain the chromatic polynomial of the following graph using the theorem,

( ) ( ) ( )n n nP of G P of G P of G . (Nov./Dec. 2017)

22. State and prove five-color theorem. (Nov./Dec. 2017) (R)

23. Define the following and give one example to each: Complete Matching, Minimal Covering,

Balanced Digraph, Strongly Connected Digraph, Fragment in a digraph. (Nov./Dec. 2017)

(A)

24. Prove that a digraph G is an Euler digraph if and only if G is connected and is balanced.

Draw an example Euler digraph of 6 vertices. (Nov./Dec. 2017) (A)

25. Define chromatic polynomial and write the chromatic polynomial of a graph with n vertices.

(Nov/Dec.2018) (U)

26. Define complete matching and minimal covering in a graph G. Give one application example

to each. (Nov/Dec.2018) (A)

27. Define the following and give one example to each. (Nov/Dec.2018) (A)

(1) Complete symmetric digraph (2) Balanced digraph

(3) Equivalence graph (4) Accessibility in a graph

28. When is a digraph an Euler digraph? Draw an Euler digraph. (Nov/Dec.2018) (R)

UNIT IV PERMUTATIONS & COMBINATIONS

1. Define fundamental principle of counting? Give an example. (R)

Defn:

The fundamental counting principle is a mathematical rule that allows you to find the

number of ways that a combination of events can occur.

For example, if the first event can occur 3 ways, the second event can occur 4 ways, and

the third event can occur 5 ways, then you can find out the number of unique

combinations by multiplying: 3 * 4 * 5 = 60 unique combinations.

2. State Binomial theorem. (R)

Statement:

For any integer 0n 1 2 2 1

0 1 2 1

0

( ) ........n n n n n nn nn

n i i

i

i

x y nC x nC x y nC x y nC x y nC y

nC x y

3. Define permutation. (R)

Defn:

A permutation of n objects taken r at a time is an arrangement (or ordered selection) of r

of the objects (r ≤ n).

4. In how many ways they can be arranged two letters at a time from the letters a, b, c? (A)

Ans:

The arrangements of the letters a, b, c taken two at a time are ab, ba, ac, ca, bc, cb.

The number of permutation of 3 letters taken two at a time =3P2 = P(3, 2) = 6 Therefore

the number of 2-arrangements are 6.

5. In how many different ways can the letters of the word “LEADING” be arranged in such a

way that the vowels always come together. (A) (Nov./Dec 2016)

6. Define Sum rule and product rule? (R)

Sum rule: Suppose a certain operation (or process) can be done in m ways and another

operation (or process) can be done in n ways and both cannot be done

simultaneously. Then either of them can be done in m+n ways.

Product rule: Suppose a certain operation (or process) can be done in m ways and another

independent of the former can be done in n ways. Then both of them can be

done in mn ways.

7. Find the total number of 9 digit numbers which have all different digits. (U)

Ans: The digits used forming the numbers are 0,1,2,3,4,5,6,7,8,9.

The highest place can be filled only with the 9 digits 1, 2, 3, 4, 5, 6, 7, 8, 9 in 9 ways. The

remaining 8 places can be filled with the remaining 9 digits including 0 in 9! Ways. So total

number of 9 digits numbers = 9.9! = 3265920.

8. A committee including 3 boys and 4 girls is to be formed from a group of 10 boys and 12

girls. How many different committees can be formed from the group? (A) (Nov./Dec 2016)

9. How many bit strings of length 10 contain exactly four 1’s? (A)

Ans: Total number of 10 bit strings with four 1’s = 10C4 = 210.

10. How many different bit strings are there of length seven? (A)

Ans: In a bit string each place can be filled by 0 or 1 (2 choices). To have a bit string of

length 7, all the 7 places can be filled independently by 0 or 1. So 27 = 128 different bit

strings will be there.

11. Find the number of arrangements of the letters in MAPPANASSRR. Also find how many of

these arrangements have no adjacent A’s? (A)

Ans: There are total 11 alphabets of which 3 A’s, 2 P’s, 2 S’s and 2 R’s.

Number of different arrangements = 831600!2!2!2!3

!11

If we tie up the 2A’s as a single unit then we have 10 units

Number of arrangements that have 2A’s adjacent = 453600!2!2!2

!10

Number of arrangements that have no 2 A’s adjacent = 836100-453600=382500

12. Find the number of arrangements of the letters in DISCRETE. How many of these

arrangements have no adjacent E’s? (A)

Ans: Number of different arrangements = 20160!2

!8

If we tie up 2 E’s as a single unit then we have 7 units

Number of distinct arrangements that have adjacent E’s = 7! =5040

Number of arrangements that don’t have adjacent E’s = 20160-5040 =15120.

13. How many ways can the letters in ENGINEERING be arranged so that all three e’s together? (R)

(Nov/Dec 2018)

14. How many permutation of {a, b, c, d, e, f, g} end with a? (A)

Ans. 6P6 = 6! = 720.

15. A label identifier, for a computer system consists of one English alphabet in capital letter

followed by two nonzero digits. If repetition of digits is allowed, how many label identifiers

are possible? (R)

Ans: For the first place there are 26 (alphabets) choices. For the second place there are 9

(digits from 1 to 9) choices. For the third place there are 9 choices (repetition allowed).

Therefore number of identifiers = 26*9*9 = 2106.

16. Prove that 2 ( , 2) ( , )P n n P n n (U)

Proof:! !

2 ( , 2) 2 != ( )( ( 2))! ( 2)!

n nP n n n P n,n

n n n n

17. How many permutations of {a, b, c, d, e, f, g} end with a? (A) Soln. This is a permutation with repeats not allowed. Additionally, the last position must be an ’a’,

so we have only 6 items to place. Therefore, there are P(6, 6) = 6!/0! = 720 permutations.

18. How many different bit strings are there of length seven? (A) Soln. In a bit string each place can be filled by 0 or 1 (2 choices).

To have a bit string of length 7, all the 7 places can be filled independently by 0 or 1. So

27 = 128 different bit strings will be there.

19. How many permutations are there in the word MISSISSIPPI? (A)

Ans. The permutations of the given word are 11!

346501!4!4!2!

20. Define combination. (R)

Defn: A combination of n objects taken r at a time is an unordered selection of r of the n

objects (r ≤ n).

21. How many ways are there to select five players from 10-membr tennis team to make a trip to

a match at another school? (U)

Ans. 10!

(10,5) 2525!5!

C .

22. Show that rnCrnCrnC ,11,1, . (U)

Ans:

rnC

rnr

n

rnr

n

rnr

n

rrnrnr

n

rnrr

n

rnrnr

n

rnr

n

rnr

nrnCrnC

,!!

!

!1!1

!1

11

!1!1

!1

!1!1

!1

!1!1

!1

!1!

!1

!!1

!1,11,1

23. Find the number of unordered samples of size five (repetition allowed) from

fedcba ,,,,, . (U)

Ans: n=6 r=5 252!5!5

!10

5

101

r

rn

24. In how many ways a football team of eleven players can be chosen out of 18 players when

(a) four particular players are to be always included (b) three particular players are to be

always excluded. (U)

Ans:

(a) 4 particular players are selected remaining 7 players need to be selected from

remaining 11 players in 330!4!7

!11

7

11

(b) Three particular players are excluded. So 11 players are selected from 15 available

players in 1365!4!11

!15

11

15

25. How many teams of six with a captain can be selected from 12 persons? (U)

Ans: Captain is selected in 12 ways. Remaining 5 people can be selected from remaining 11

persons in 462!6!5

!11

5

11

.

26. There are 6 multiple choice questions is an examination. If the first 3 questions have 4

choices each and the next three questions have 5 choices, how many sequences of answers

are possible? (U)

Ans: The first 3 questions can be answered in 4 x 4 x 4 = 43 ways (Product rule). Similarly,

the next 3 questions can be answered in 5 x 5 x 5 = 53 ways (Product rule). Therefore

total number of ways answering all the 6 questions = 43 x 5

3 = 800.

27. What is the coefficient of 12 13

x y in the expansion of 25

( )x y ? (U)

Ans. Using Binomial theorem it follows that this coefficient is

13

25!25 5200300

13!12!C

28. Find the coefficient of 60x in 8 9 10 7....)(x x x . (U)

Ans. 87

78 9 10 7 7 2 7 56 561....) ) (1 ......) 11

( (x x x x x x x x xx

Then the coefficient of 60x in 8 9 10 7....)(x x x is the coefficient of 4x in 71 x ,

which is 4 4 44 4 4

7 ( 1) ( 1) (7 4 1 )( 1) 10C C C

29. Prove that 0

2n

n

kk

nc

, where n be a nonnegative integer. (U)

Proof. Using Binomial theorem, with 1, 1x y , we write

0 0

1 1 1 12 ( )n n

n n k n k

k kk k

nc nc

Hence proved.

30. Find the expansion of 4

( )x y and6

( )x y . (U)

31. How many ways are there to place ten indistinguishable balls into eight distinguishable bins?

Ans. The number of ways to place ten indistinguishable balls into eight distinguishable

bins equals the number of 10 combinations from a set with eight elements when repetition is

allowed. Consequently, there are

17!

(8 10 1,10) (17,10) 1944810!7!

C C ways.

32. State the principle of inclusion and exclusion. (R)

Statement: In combinatorics, the inclusion-exclusion principle is an equation relating the size

of two sets and their union. If A1, A2, A3,……, An are n finite sets,

.

33. How many positive integers not exceeding 1000 are divisible by 7 or 11? (U) Soln. Let A and B be sets of positive integers divisible by 7and 11respectively.

No. of integers divisible by 7 or 11 is |A ∪B| = |A| + |B| – |A∩B|

1000 1000 1000

7 11 7 11

142 90 12 220

34. Determine the number of positive integers n,1 n 500 , that are not divisible by 5 or 5.

(Nov/Dec 2018) (U)

35. What is derangement of a set? (U)

Ans. A derangement of a set A is a bijection from A into itself that has no fixed points. It is a

well-known application of the inclusion–exclusion principle is to the combinatorial

problem of counting all derangements of a finite set.

36. Give any example for derangement of 12345? (U)

Ans. The permutation 21453 is derangement of 12345, since no number is left in its original

position.

37. Find the number of derangements of 1, 2, 3, 4. List all the derangements. (U)

38. There are four balls of different colors and four boxes of colors same as those of the balls.

Find the number of ways in which the balls, one in each box, could be placed such that a ball

does not go into a box of its own color. (U)

Ans.

The number of ways

waysD

D

9

!4

1

!3

1

!2

1

!1

11!4

4

4

39. THALASSEMIA is a genetic blood order. How many ways can the letters in THALASSEMIA be arranged so that all three A’s together? (Nov/Dec .2017)

Solution:

Number of ways of arranging all 3 A’s together 11!

2!3!.

40. Determine the number of positive numbers ,1 100n n , that are not divisible by 3 or 7. (Nov/Dec

.2017) (U)

Solution:

Given C1: n divisible by 3, C2: n divisible by 7,

1 2 0 1 2

0 1 1 2

2 1 2

1 2

( )

100, ( ) ( ) 33 14 47

( ) 4

( ) 57

N C C S S S

S N S N C N C

S N C C

N C C

https://en.wikipedia.org/wiki/Bijectionhttps://en.wikipedia.org/wiki/Derangement

PART B

1. How many positive integers n can be formed using the digits 7,6,5,5,4,4,3 if n has to

exceed 5000000? (A)

2. A box contains six white balls and five red balls. Find the number of ways four balls can be

drawn from the box if (1) they can be any colour (2) two must be white and two red (3) they

must all be the same colour. (U)

3. How many arrangements are there of all the vowels adjacent in SOCIOLOGICAL?

(U) (Nov./Dec 2016)

4. Find the value of n for the following 2 ( ,2) 50 (2 ,2)P n P n . (R) (Nov./Dec 2016)

5. How many distinct four-digit integers can one make from the digits 1, 3, 3, 7, 7 and 8?

(A) (Nov./Dec

2016)

6. In how many possible ways could a student answer a 10-question true-false test?

(A)

(Nov./Dec 2016)

7. A bit is either 0 or 1. A byte is a sequence of 8 bits. Find the number of bytes. Among these

how many are (1) starting with 11 and ending with 00 (2) starting with 11 but not ending

with 00 or not starting with 11 but ending with 00?

8. How many arrangements of the letters in MISSISSIPPI has no consecutive S’s?

(U)

(Nov./Dec 2016)

9. A gym coach must select 11 seniors to play on a football team. If he can make his selection

in 12,376 ways, how many seniors are eligible to play? (A) (Nov./Dec 2016)

10. How many permutations of size 3 can one produce with the letters m, r, a, f and t?

(A) (Nov./Dec

2016)

11. Rama has two dozen each of n different colored beads. If she can select 20 beads (with

repetitions of colors allowed), in 230,230 ways, what is the value of n? (A) (Nov./Dec 2016)

12. A question paper has 3 parts, Part A , Part B and Part C having12 , 4 and 4 questions

respectively. A student has to answer 10 questions from Part A and 5 questions from Part B

and Part C put together selecting atleast 2 from each one of these two parts. In how many

ways the selection of questions can be done? (A)

13. A computer password consists of a letter of English alphabet followed by 2 or 3 digits. Find

the following (1) the total number of passwords that can be formed (2) the number of

passwords in which no digit repeats. (A)

14. How many bit strings of length 12 contain (i) exactly three ones? (ii) atmost three ones? (iii)

at least three ones? (iv)An equal number of zeros and ones? (A)

15. Five boys and five girls are to be seated in row. In how many ways can be they be seated if

(i) the boys are together and the girls are together (ii) no two girls are together (iii) The boys

and girls alternate. (A)

16. In how many ways can 6 students and 4 teachers be arranged in a row for a photograph if (i)

no two teachers are together (ii) the teachers are all together? (A)

17. Show that if n and k are positive integers then1

( 1) /1

n nn k

k k

. (U)

18. Let n and k be positive integers with n ≥ k. Prove that c(n+1, k) = c(n, k-1) + c(n, k) . (U) 19. Prove that the number of different permutations of n objects, where there are n1

indistinguishable objects of type 1, n2indistinguishable objects of type 2,…., and nk

indistinguishable objects of type k, is

1 2

!

! !...... !k

n

n n n. (U)

20. Determine the number of positive integers 10001, nn that are not divisible by 2, 3 or 5.

(A)

21. Determine the number of integers between 1 and 200 that are divisible by any of the integers 2, 3, 5 and 7. (A)

22. Show that if any 11 numbers are chosen from the set {1, 2,….., 20} then one of them will be a multiple of another. (U)

23. Find the number of positive integers between 1 and 250 that are not divisible by any one of the integers 2, 3, 5 and 6. (A)

24. Show that in any set of eleven integers, there are two integers whose difference is divisible by 10. (U)

25. How many solutions does the equations 11321 xxx have, where 0, 321 xandxx such

that 4,3 21 xx and 63 x ? Use the principle of inclusion-exclusion. (A)

26. There are 2500 students in a college, of these 1700have taken a course in C , 1000have

taken a course Pascal and 550 have taken a course in Networking. Further 750 have taken

courses in both C and Pascal, 400 have taken courses in both C and Networking and 275

have taken courses in both Pascal and Networking. If 200 of these students have taken

courses in C , Pascal and Networking. (A)

(1) How many of these 2500 students have taken a course in any of these three courses C ,

Pascal and Networking?

(2) How many of these 2500 students have not taken a course in any of these three courses

C , Pascal and Networking?

27. There are 2500 students in an engineering college. Of these 188have taken a course in

FORTRAN, 100 have taken a course in C and 35 have taken a course in Java. Further 88

have taken a course in both FORTRAN and C , 23 have taken a course in both C and Java

and 29 have taken a course in both FORTRAN and Java. If 19 of these students have taken

all of these three courses, how many of these 2500students have not taken a course in any of

these three courses? (A)

28. How many derangements of 1, 2,3, 4,5, 6 begin with 1, 2, and 3 in some order? and end with 1, 2, and 3 in some order. (A)

29. In how many ways we can arrange the numbers 1, 2, 3,…… 10 so that 1 is not in the 1stplace, 2 is not in the 2

nd place and so on, and 10 is not in the 10

th place? (A)

30. Show that the number of derangements of a set with n elements is (U)

1 1 1 1

1 ...... ( 1)1! 2! 3! !

!nn

Dn

n

.

31. There are five students in a group and their roll numbers are, S1, S2, S3, S4, S5 and S6. They

are given with five assignments numbered 1 to 6. Each has to solve one assignment. How

many ways the arrangements can be distributed such that a student is not getting assignment

number same as his roll number? (Nov/Dec .2017)

32. Determine the coefficient of 9 6x y in the expansion of 15(4 )y x . (Nov/Dec .2017)

33. How many integer solutions for 1 2 3 4 5 40x x x x x where 3, 1 5ix i . (Nov/Dec

.2017)

34. In a survey of chewing gum tastes of a group of baseball players, it was found that 22 liked juicy

fruit, 25 liked spearmint, 39 liked bubble gum, 9 liked both spearmint and juicy fruit, 17 liked juicy

fruit and bubble gum, 20 liked spearmint and bubble gum, 6 liked all three and 4 liked none of these.

(Nov/Dec .2017) (A)

35. Determine the numbr of six integers (no leading zeros) in which

(i) No digit may be repeated (ii) Digits may be repeated. Answer parts

(i) and (ii) with extra condition that the six digit number even: Also do the same with the

condition that the number is divisible by 5. (Nov/Dec .2018) (R)

At a nursery, Reshmi wants to arrange 15 different plants on five shelves for a window

display. In how many ways can she arrange them so that each shelf has at least one, but no

more than four plants? (Nov/Dec .2018) (A)

UNIT V GENERATING FUNCTIONS

Part - A

1. Define partition of integer. (R)

Defn. A partition of a positive integer n is a multiset of positive integers that sum to n.

We denote the number of partitions of n by pn.

2. Define exponential generating function. (R)

Defn. For a sequence 0 1 2 3, , , ,.....a a a a of real numbers

0 1 2 3

2 3

0

( ) .........2! 3! !

n

n

n

x x xf x a a x a a a

n

.

3. Find the generating function for the sequence 8, 26, 54, 92,…… (U) 4. Define recurrence relation.(R) (Nov./Dec 2016) 5. Define generating function. Give an example to a polynomial and a power series.

(R) (Nov./Dec 2016,

2018)

6. Write a homogeneous and non- homogeneous recurrence relation. (R) (Nov./Dec 2018)

7. Find the coefficient of 6 8(3 5 )x in x . (Nov./Dec 2017)

Solution:

88 8

0

6

6 8 66 614

5(3 5 ) (3) (7 )

3

5 1(3) (7 6) 13 5

3 3

r

rr

rx r C x

Coefficient of x C C

8. The number of virus affected files in a system is 500 (approximately) and this doubles every four hours. Using a recurrence relation, determine the number of virus affected files in the

system after one day. (Nov./Dec 2017)

1 1

40

2 3

th

n

n n n n n

a number of virus at hour

a a a a a

The number of virus present after one day

6

6 500 (3) 364,500a .

Part - B

1. Show that the number of partitions of a positive integer n, where no summand appears more

than twice equals the number of partitions of n, where no summand is divisible by 3. (U)

2. Determine the coefficient of 8x in

2

1

( 3)( 2)x x . (A)

3. Determine the generating function for the numeric function 2

2

r

r r

if r is evena

if r is odd

(A)

4. Using generating function, find the number of partitions of n = 6. (A)

5. Find the exponential generating function for each of the following sequences. (A)

(i) 1, -1, 1, -1, 1, -1,….

(ii) 1, 2, 22, 23, 24,….

(iii) a, a3, a5, a7,….. a € R.

6. Using exponential generating function, find the number of ways in which 4 of the letters in

the words given below be arranged: (i) ENGINE (ii) HAWAII (A)

7. Determine the sequences generated by each of the following exponential generating

functions. (A)

(i) 3( ) 3 xf x e , (ii)

2( ) xf x e x (iii) ( ) 3/(1 2 )xf x x e

8. A ship carries 48 flags, 12 each of the colors red, white, blue and black. Twelve of these flags

are placed on a vertical pole in order to communicate a signal to other ships. (A)

(i) How many of these signals use an even number of blue flags and an odd number of

black flags?

(ii) How many of the signals have atleast three white flags or no white flags at all?

9. Find the formula to express: 02 + 12 + 22 + ……..+n2 as a function of n using summation on

operator. (A)

10. Find the recurrence relation and the initial condition for the sequence 2, 10, 50, 250,…..

Hence find the general term of the sequence. (A)

11. Solve the recurrence relation 3,0,3 02

1 annnaa nn . (A)

12. Solve 4 1 4 2 3 2 , 0 1, 1 1kS k S k S k k S S . (A)

13. Solve the recurrence relation 2,1,2,2 1021 aanwhereaaa nnn .(A)

14. Solve the recurrence relation 065 21 nnn aaa , where 5,2 10 aa . (A)

15. Solve the recurrence relation n

nnn aaa 386 21 for 2n where 73 10 aa . (A)

16. Write the recurrence relation for Fibonacci numbers and hence solve it. (U)

17. Solve the recurrence relation of the Fibonacci sequence of numbers 2,21 nfff nnn

with the initial conditions 1,1 21 ff . (A)

18. Solve 7 1 10 2 6 8Y n Y n Y n n with 0 1, 1 2Y Y . (A)

19. Solve the recurrence relation 06116 123 nnnn yyyy with 0,2 10 yy and 22 y .

(A)

20. Find a formula for the general term nF of the Fibonacci sequence ....,13,8,5,3,2,1,1,0 . (U)

21. Solve 3 1 4 2 4kS k S k S k . (A)

22. Solve 1 02 2 , 2n

n na a a . (A)

23. Use generating function to determine how many four element subjects of

S = {1, 2, 3, ….,15} contain no consecutive numbers? (A)

24. Using the generating function, solve the difference equation 0612 nnn yyy ,

2,1 01 yy .

25. Using generating function solve : 0,065 12 nyyy nnn with 1,1 10 yy . (A)

26. Use the method of generating function to solve the recurrence relation

2;444 21 naaan

nnn given that 8,2 10 aa . (A)

27. Find the generating function of Fibonacci sequence. (U)

28. Find the generating function of Fibonacci sequence 2,21 nnFnFnF with

110 FF . (A)

29. Using generating function solve 1,1,3 01 annaa nn . (A)

30. Solve the following recurrence relation n

nnn aaa 22 12 with initial conditions

1,2 10 aa by using generating functions.(A)

31. Solve the recurrence relation using the method of generating function

0,40034 12 naaa nnn 330,300 10 aa . (A)

32. Using generating functions, solve the recurrence relation 0158 12 nnn aaa given that

8,2 10 aa . (A)

33. Solve 2,02312 nnSnSnS with 11,30 SS by using generating function. (A)

34. Use the method of generating function to solve the recurrence relation

1,411681 nnSnSnS n with 10 S and 81 S . (A)

35. Solve by using generating function, the recurrence relation n

nn yy 421 with 10 y for

0n . (A)

36. Find the generating function of the recurrence relation kaa kk 295 1 with 60 a and

hence solve it. (A)

37. Use the method of generating function to solve 1,1,13 01 anaa nn . (A)

38. Discuss about exponential generating function with an example. (A) (Nov./Dec 2016)

39. Find the unique solution of the recurrence relation 343,1,07 31 anaa nn .

(A) (Nov./Dec

2016)

40. The population of Mumbai city is 6,000,000 at the end of the year 2015. The number of

immigrants is 20000 n at the end of the year n. The population of the city increases at the rate

of 5% per year. Use a recurrence relation to determine the population of the city at the end of

2025.

(A)

(Nov./Dec 2016)

41. Write short notes on summation operator. (A) (Nov./Dec 2016)

42. Two cases of soft drinks, 24 bottles of one type and 24 bottles of another, are distributed

among five surveyors who are conducting taste tests. In how many ways can the 48 bottles be

distributed so that each surveyor gets at least two bottles of each type? And in how many

ways can they be distributed so that each surveyor gets at least two bottles of one type and

three of other type? Use generating function. (Nov./Dec 2017)

43. Find all partitions of integer 6 and find the number of partitions with distinct summands.

(Nov./Dec

2017)

44. A person invests Rs. 50,000 at 6% interest compounded annually. (Nov/Dec 2017)

(1) Find the amount at the end of 1st, 2nd, 3rd year.

(2) Write the general explicit formula.

(3) How long will it take to double the investment? Use recurrence relation.

45. Derive an explicit formula for the Fibonacci sequence using recurrence relation.

(A) (Nov./Dec

2017)

46. What is Ferrer’s graph? Give an example for Ferrer’s graph and its transposition graph.

(Nov./Dec

2018) (R)

47. Explain exponential generating function with an example. (Nov./Dec 2018) (R)

48. Explain the summation operator with an example. (Nov./Dec 2018) (R)

49. What is meant by recurrence relation? Write one application of each first order and second

linear homogeneous recurrence relation with examples. (Nov./Dec 2018) (R)