8/10/2019 a4[1] Revised

1/2

CS201 : Discrete Mathematics

Semester II, 2010-11, CSE, IIT Kanpur

Assignment - 4 (due on 18th April)

Note : Give complete details of the analysis of your solution.

1. Note that the group isomorphism can be extended naturally to ring isomorphism as follows.

Definition: An isomorphism of a ringR,+, with a ringR,,is a one-to-one and

onto function mappingR onto R such that for all a, b R,

(a+b) =(a) (b)

(a b) =(a) (b)

Prove that

nZ,+, is a ring under usual addition and multiplication.

2Z,+, is not isomorphic to 3Z,+, .

2. In the class on 18th March, we gave a brief introduction of an algorithmic process to prove

some results in graph theory. Use a reasoning involving such an algorithmic process to prove

that every graph onn vertices and m edges has a subgraph where minimum degree is at least

(mn).

3. girthof a graph is the length of the smallest cycle in the graph. Let k be any positive integer.

(a) For a graph on n vertices where degree of each vertex is at least n1/k, prove that girth

is O(k).

(b) Use the previous exercise to conclude that every graph withn1+1/k edges has girthO(k).

4. In one of the earlier lectures on graph theory, we gave an upper bound on the number of

vertices in terms of the radius and maximum degree of the graph. In this exercise, we wish

to design a lower bound on the number of vertices in terms of some parameters.

Derive the best possible lower bound on the number of vertices of a graph in terms of its girth

g and minimum degree d.

5. As discussed in the class, there are two ways to represent/store a graph. These are Adjacency

lists and Adjacency matrix. Adjacency matrix occupies (n2) space which is too large com-

pared to O(m) space occupied by Adjacency list representation. Moreover, for most of the

graph algorithmic problems, the use of Adjacency matrix imposes an overhead of (n2) on

the running time of the algorithm. However, there are some exception. One such exception

is Detecting a sink in a graph problem (discussed on 18th March). Here is another very

interesting but challenging exception:

1

8/10/2019 a4[1] Revised

2/2

A graph on n vertices is a scorpio graph if it has a vertex of degree 1(the sting) connected

to a vertex of degree two (the tail) connected a vertex of degree n 2 (the body) connected

to the other n 3 (the feet). Each foot may be connected to some other feet but it is not

necessary. Design an algorithm that decides whether a given adjacency matrix represents a

scorpion by examining only O(n) entries. An example of scorpio is given in Figure 1.

tail

sting

body

feet

Figure 1: Scorpio graph

6. (a) Consider a n n chess board. For what value of n, is it possible to find a knights

route around the board which uses every possible move just once (in one direction or

the other)?

(b) Show that if a graph has n vertices and more than 12

(n2 3n+ 4) edges then it is

hamiltonian.

7. (a) Use Eulers formula to prove that ifG is a connected planar graph with girth 5, then

m 5(n 2)/3. Use it to deduce that the graph shown in Figure 2 (also called Petersen

graph) is not planar.

Figure 2: Petersen graph

(b) Obtain an inequality, generalizing that in part (a), for connected planar graph of girthr.

8. (a) Prove that(G) +(G) n+ 1, where graph Gis complement of graph G.

(b) Show by induction on n that the graph described in Figure 3 with 2n vertices has

chromatic polynomial k(k 1)(k2 3k+ 3)n1.

Figure 3: a ladder like Graph with 2nvertices

2