IGNOU MMTE 1 SOLVED ASSIGNMENT 2025

Ques 1.

State whether the following statements are true or false. Justify your answers with a short proof or a counterexamp

i) There exists an 8-vertex graph with three vertices of degree 3, four vertices of degree 2 and one vertex of degree 1

ii) The neighbour of every leaf is a cut-vertex.

iii) Every line graph of a bipartite graph is 2-colourable

iv) is a graphic sequence then so is 

v)

vi) A Hamiltonian graph has no cut-vertices.

vii) The Petersen graph is 3-critical.

viii) An n-vertex star has no perfect matching for n ≥ 3.

ix) The crossing number of K_3,3 is 2.

x) If f and g are two flows on a network N, then max is also a flow.

Ques 2.

2. (a) If every cycle in a graph is even, then prove that the graph is bipartite. Is its converse true. Prove or disprove.

Ques 3.

(b) For each n-vertex h-level complete binary tree, prove that 

Ques 4.

(c) Check whether the following graphs G and H are isomorphic or not.

Ques 5.

(a) Prove or disprove: A connected graph with order and size equal must contain exactly one cycle.

Ques 6.

Check whether the following graphs G and H are isomorphic or not.

Ques 7.

(a) Prove or disprove: A connected graph with order and size equal must contain exactly one cycle.

Ques 8.

(b) Find a minimum-weight spanning tree in the following graph.

Ques 9.

(c) Determine the number of non-planar graphs with 6 vertices. 

Ques 10.

(d) Find the chromatic and edge-chromatic numbers of the following graph

Ques 11.

(a) Show that there are 14 spanning trees of the following graph. Draw all the spanning trees.

Ques 12.

Is it possible that a graph is 3-chromatic but not 3-critical? If so, explain it with an example.

Ques 13.

(c) Check the sequence (6, 5, 4, 4, 3, 1, 1, 1, 1) is graphic or not. Also, find a graph realising it.

Ques 14.

(a) Verify Euler’s formula for the following plane graph.

Ques 15.

(b) Check whether the graph is planar or not.

Ques 16.

(c) For every graph True or false? Justify.

Ques 17.

(d) Find the matching number of the line graph of the graph given in part(a).

Ques 18.

(a) What is the maximum possible flow that can pass through the following network N? Define such a flow

Ques 19.

Show that [S, T] is an (s, t)-cut in network N give in part(a),where Does N have an other (s, t)-cut with capacity smaller than Cap(S, T)? What is the maximum possible value of a flow in N?

Ques 20.

(c) State and prove Hall’s Theorem

Ques 21.

(d) Provide an example of a 3-regular planar graph with 8-vertices. Is this graph a maximal planar graph? Why?

Ques 22.

(a) Find the values of n and m for which the star graph S_n,mis Eulerian.

Ques 23.

(b) Using Fleury’s algorithm, find an Eulerian circuit in the following graph.

Ques 24.

(c) Prove or disprove: If G is a graph with χ(G) denoting its chromatic number, then

Ques 25.

(a) Find the line graph of the following graph? Write number of vertices and edges in the line graph.

Ques 26.

Find the thickness and crossing number of the graph G given in Q.2(c)?

Ques 27.

(c) Drawwith explanation.

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