(a) Let G be a simple graph with n vertices and m edges where m ≥ (^₂¹) + 2. (i) Let vand w be any two non-adjacent vertices. Explain why the graph G\{v, w} has at most (2) edges (where (8) is the binomial coefficient). a! = (a-b)!!b! (ii) Is G is necessarily Hamiltonian? Justify your answer.

(a) Let G be a simple graph with n vertices and m edges where m ≥ (^₂¹) + 2. (i) Let vand w be any two non-adjacent vertices. Explain why the graph G\{v, w} has at most (2) edges (where (8) is the binomial coefficient). a! = (a-b)!!b! (ii) Is G is necessarily Hamiltonian? Justify your answer.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

(a) Let G be a simple graph with n vertices and m edges where m ≥ (^≥¹) + 2.
(i) Let v and w be any two non-adjacent vertices. Explain why the graph
G\{v, w} has at most (z2) edges (where (1)
is the binomial
coefficient).
a!
(a-b)!!b!
(ii) Is G is necessarily Hamiltonian? Justify your answer.
=
(b) Is it necessarily the case that for a graph with n vertices and exactly (₂¹)+1
edges be Hamiltonian? Either show this must be true of provide a counterex-
ample.

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