IGNOU MMTE 1 SOLVED ASSIGNMENT

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.

Ques 28.

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 29.

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 30.

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

Ques 31.

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

Ques 32.

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

Ques 33.

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

Ques 34.

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

Ques 35.

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

Ques 36.

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

Ques 37.

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

Ques 38.

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

Ques 39.

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

Ques 40.

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

Ques 41.

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

Ques 42.

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

Ques 43.

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

Ques 44.

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

Ques 45.

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

Ques 46.

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 47.

(c) State and prove Hall’s Theorem

Ques 48.

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

Ques 49.

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

Ques 50.

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

Ques 51.

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

Ques 52.

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

Ques 53.

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

Ques 54.

(c) Drawwith explanation.

Ques 55.

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 56.

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 57.

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

Ques 58.

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

Ques 59.

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

Ques 60.

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

Ques 61.

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

Ques 62.

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

Ques 63.

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

Ques 64.

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

Ques 65.

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

Ques 66.

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

Ques 67.

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

Ques 68.

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

Ques 69.

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

Ques 70.

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

Ques 71.

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

Ques 72.

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

Ques 73.

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 74.

(c) State and prove Hall’s Theorem

Ques 75.

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

Ques 76.

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

Ques 77.

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

Ques 78.

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

Ques 79.

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

Ques 80.

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

Ques 81.

(c) Drawwith explanation.

Ques 82.

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 83.

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 84.

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

Ques 85.

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

Ques 86.

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

Ques 87.

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

Ques 88.

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

Ques 89.

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

Ques 90.

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

Ques 91.

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

Ques 92.

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

Ques 93.

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

Ques 94.

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

Ques 95.

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

Ques 96.

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

Ques 97.

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

Ques 98.

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

Ques 99.

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

Ques 100.

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 101.

(c) State and prove Hall’s Theorem

Ques 102.

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

Ques 103.

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

Ques 104.

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

Ques 105.

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

Ques 106.

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

Ques 107.

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

Ques 108.

(c) Drawwith explanation.

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