IGNOU MMTE001 Solved Assignment 2024  M.Sc. MACS
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IGNOU MMTE001 Assignment Question Paper 2024
mmte001solvedassignment2024bc500e5856224aaebac575e86f0373be
 State whether the following statements are true or false. Justify your answers with a short proof or a counterexample
i) There exists no 9vertex graph with three vertices of degree 3 , four vertices of degree 2 and two vertices of degree 1 .
ii)${C}_{6}\vee {P}_{4}$ ${C}_{6}\vee {P}_{4}$ C_(6)vvP_(4) C_6 \vee P_4 has a cycle of length at least 7 .${C}_{6}\vee {P}_{4}$
iii) The diameter of a graph cannot exceed its girth.
iv) Every Hamiltonian graph is Eulerian.
v) Every 3connected graph is 3edgeconnected.
vi) If$G$ $G$ G G is an Eulerian graph, then so is$G$ $L(G)$ $L(G)$ L(G) L(G) .$L(G)$
vii)${\chi}^{\mathrm{\prime}}({C}_{3}\times {K}_{2})=3$ ${\chi}^{\mathrm{\prime}}\left({C}_{3}\times {K}_{2}\right)=3$ chi^(‘)(C_(3)xxK_(2))=3 \chi^{\prime}\left(C_3 \times K_2\right)=3 .${\chi}^{\mathrm{\prime}}({C}_{3}\times {K}_{2})=3$
viii) There exists no graph$G$ $G$ G G with$G$ $\chi (G)>\omega (G)+1$ $\chi (G)>\omega (G)+1$ chi(G) > omega(G)+1 \chi(G)>\omega(G)+1 .$\chi (G)>\omega (G)+1$
ix) The minimum size of a$k$ $k$ k k chromatic graph is$k$ $\left(\begin{array}{c}k\\ 2\end{array}\right)$ $\left(\begin{array}{c}k\\ 2\end{array}\right)$ ([k],[2]) \left(\begin{array}{c}k \\ 2\end{array}\right) .$\left(\begin{array}{c}k\\ 2\end{array}\right)$
x) The 6dimensional hypercube${Q}_{6}$ ${Q}_{6}$ Q_(6) Q_6 has no perfect matching.${Q}_{6}$  (a) The complement of the Petersen graph is 2connected. Prove or disprove.
(b) Consider a graph$G$ $G$ G G . Let$G$ $x,y\in V(G)$ $x,y\in V(G)$ x,y in V(G) x, y \in V(G) be such that$x,y\in V(G)$ $x\leftrightarrow y$ $x\leftrightarrow y$ x harr y x \leftrightarrow y . Show that for all$x\leftrightarrow y$ $z\in V(G),d(x,z)d(y,z)\le 1$ $z\in V(G),d(x,z)d(y,z)\le 1$ z in V(G),d(x,z)d(y,z) <= 1 z \in V(G),d(x, z)d(y, z) \leq 1 .$z\in V(G),d(x,z)d(y,z)\le 1$
(c) Check whether the following graphs$G$ $G$ G G and$G$ $H$ $H$ H H are isomorphic or not.$H$
 (a) Let
$G$ $G$ G G be a connected$G$ $n$ $n$ n n vertex graph. Prove that$n$ $G$ $G$ G G has exactly one cycle iff$G$ $G$ $G$ G G has exactly$G$ $n$ $n$ n n edges.$n$
(b) Find a minimumweigh spanning tree in the following graph.
(d) Find the chromatic and edgechromatic numbers of the following graph.
 (a) Find the number of spanning trees of the following graph.
(c) Give an example of a 4critical graph different from a complete graph. Justify the choice of your example.
(d) State and prove the Handshaking Lemma for planar graphs.
 (a) Verify Euler’s formula for the following plane graph.
(c) What is the minimum possible thickness of a 4connected trianglefree graph on 8 vertices? Also draw such a graph.
(d) Define the parameters
6. (a) What is the maximum possible flow that can pass through the following network? Define such a flow.
(c) Let
 (a) Find the values of
$n$ $n$ n n for which$n$ ${Q}_{n}$ ${Q}_{n}$ Q_(n) Q_n is Eulerian.${Q}_{n}$
(b) Using Fleury’s algorithm, find an Eulerian circuit in the following graph.
MMTE001 Sample Solution 2024
mmte001solvedassignment2024ss8e24e61006c94b4384f6a5bf6ef5ab5c
 State whether the following statements are true or false. Justify your answers with a short proof or a counterexample
i) There exists no 9vertex graph with three vertices of degree 3 , four vertices of degree 2 and two vertices of degree 1 .
 Three vertices of degree 3
 Four vertices of degree 2
 Two vertices of degree 1
${C}_{6}$ ${C}_{6}$ C_(6) C_6 represents a cycle graph with 6 vertices, which means it has a cycle of length 6.${C}_{6}$ ${P}_{4}$ ${P}_{4}$ P_(4) P_4 represents a path graph with 4 vertices, which is a linear graph with no cycles.${P}_{4}$  The operation
$\vee $ $\vee $ vv \vee represents the join of two graphs, which means connecting every vertex of the first graph to every vertex of the second graph.$\vee $
 The diameter of a graph is the length of the longest shortest path between any two vertices in the graph.
 The girth of a graph is the length of the shortest cycle in the graph.
 A Hamiltonian graph is a graph that contains a Hamiltonian cycle, which is a cycle that visits every vertex exactly once.
 An Eulerian graph is a graph in which all vertices have even degree, and there exists a cycle (called an Eulerian cycle) that traverses every edge exactly once.
 A graph is said to be
$k$ $k$ k k connected if there is no set of$k$ $k1$ $k1$ k1 k1 vertices whose removal disconnects the graph.$k1$  A graph is said to be
$k$ $k$ k k edgeconnected if there is no set of$k$ $k1$ $k1$ k1 k1 edges whose removal disconnects the graph.$k1$
 An Eulerian graph is a graph in which there exists a closed walk that visits every edge exactly once.
$L(G)$ $L(G)$ L(G) L(G) represents the line graph of$L(G)$ $G$ $G$ G G , which is a graph whose vertices represent the edges of$G$ $G$ $G$ G G and where two vertices in$G$ $L(G)$ $L(G)$ L(G) L(G) are adjacent if and only if their corresponding edges in$L(G)$ $G$ $G$ G G are incident (share a common vertex) in$G$ $G$ $G$ G G .$G$
${\chi}^{\mathrm{\prime}}(G)$ ${\chi}^{\mathrm{\prime}}(G)$ chi^(‘)(G) \chi^{\prime}(G) represents the edge chromatic number of a graph${\chi}^{\mathrm{\prime}}(G)$ $G$ $G$ G G , which is the minimum number of colors needed to color the edges of$G$ $G$ $G$ G G such that no two adjacent edges share the same color.$G$ ${C}_{3}$ ${C}_{3}$ C_(3) C_3 is the cycle graph with 3 vertices, also known as a triangle.${C}_{3}$ ${K}_{2}$ ${K}_{2}$ K_(2) K_2 is the complete graph with 2 vertices, which is essentially a single edge.${K}_{2}$ ${C}_{3}\times {K}_{2}$ ${C}_{3}\times {K}_{2}$ C_(3)xxK_(2) C_3 \times K_2 represents the Cartesian product of${C}_{3}\times {K}_{2}$ ${C}_{3}$ ${C}_{3}$ C_(3) C_3 and${C}_{3}$ ${K}_{2}$ ${K}_{2}$ K_(2) K_2 , which results in a graph where each vertex of${K}_{2}$ ${C}_{3}$ ${C}_{3}$ C_(3) C_3 is connected to each vertex of${C}_{3}$ ${K}_{2}$ ${K}_{2}$ K_(2) K_2 , forming a hexagon.${K}_{2}$
$\chi (G)$ $\chi (G)$ chi(G) \chi(G) represents the chromatic number of a graph$\chi (G)$ $G$ $G$ G G , which is the minimum number of colors needed to color the vertices of$G$ $G$ $G$ G G such that no two adjacent vertices share the same color.$G$ $\omega (G)$ $\omega (G)$ omega(G) \omega(G) represents the clique number of$\omega (G)$ $G$ $G$ G G , which is the size of the largest complete subgraph (clique) in$G$ $G$ $G$ G G .$G$
 A
$k$ $k$ k k chromatic graph is a graph whose chromatic number$k$ $\chi (G)$ $\chi (G)$ chi(G) \chi(G) is equal to$\chi (G)$ $k$ $k$ k k .$k$  The size of a graph refers to the number of edges in the graph.
$\left(\begin{array}{c}k\\ 2\end{array}\right)$ $\left(\begin{array}{c}k\\ 2\end{array}\right)$ ([k],[2]) \left(\begin{array}{c}k \\ 2\end{array}\right) is the binomial coefficient, representing the number of ways to choose 2 elements from a set of$\left(\begin{array}{c}k\\ 2\end{array}\right)$ $k$ $k$ k k elements, which is also equal to$k$ $\frac{k(k1)}{2}$ $\frac{k(k1)}{2}$ (k(k1))/(2) \frac{k(k1)}{2} .$\frac{k(k1)}{2}$
 A hypercube
${Q}_{n}$ ${Q}_{n}$ Q_(n) Q_n is a graph that represents an${Q}_{n}$ $n$ $n$ n n dimensional cube. It has$n$ ${2}^{n}$ ${2}^{n}$ 2^(n) 2^n vertices, each corresponding to a unique${2}^{n}$ $n$ $n$ n n bit binary string. Two vertices are adjacent if and only if their binary strings differ in exactly one bit.$n$  A perfect matching in a graph is a set of edges such that every vertex in the graph is incident to exactly one edge in the set.
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