(b) Check whether the graph is planar or not.
Short answer: Not planar. Proof (manual): Let $C_5=\{a_1,a_2,a_3,a_4,a_5\}$ and let $K_5=\{b_1,b_2,b_3,b_4,b_5\}$. Vertices of the Cartesian product $C_5\times K_5$ are pairs ______ ___ __________ _______ ___ _______ ________ _________ _________ ____ ________.
_____ ________ __________ ___ __________ _______ _________ ___ __________ ___.
_________ _________ __________ ____ ____ __________.
____ ___ _____ ______ __________.
_____ ____ ____ _____ _______ ____.
_____ _______ ____ ____ ___.
_______ ___ ______ __________ ___ ________ ________ _____ _____ ______ _____.
__________ ____ ____ ________ ________ __________ ___ ____ __________ ____.
_______ _________ _______ _____ ___.
_____ ___ ____ _________ ________ __________ _________ _________ _________.
__________ ________ _______ ______ ________ ____ _______ ______.
______ ______ ____ ______ ___ ________ _______ _____ ___ _________ _____ ___.
____ ___ ___ _____ _________ _______ _______.
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Is it possible that a graph is 3-chromatic but not 3-critical? If so, explain it with an example.
(a) What is the maximum possible flow that can pass through the following network N? Define such a flow
(c) For every graph True or false? Justify.
(c) Check whether the following graphs G and H are isomorphic or not.
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 K3,3 is 2.
x) If f and g are two flows on a network N, then max is also a flow.
(c) State and prove Hall’s Theorem
(c) Drawwith explanation.