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# UGC-NET | UGC NET CS 2015 Jun – II | Question 5

Consider a Hamiltonian Graph (G) with no loops and parallel edges. Which of the following is true with respect to this Graph (G) ?
(a) deg (v) ≥ n / 2 for each vertex of G
(b) |E(G)| ≥ 1 / 2 (n – 1) (n – 2) + 2 edges
(c) deg (v) + deg (w) ≥ n for every n and v not connected by an edge.

(A) (a) and (b)
(B) (b) and (c)
(C) (a) and (c)
(D) (a), (b) and (c)

Answer: (C)

Explanation: In an Hamiltonian Graph (G) with no loops and parallel edges:
According to Dirac’s theorem in a n vertex graph, deg (v) ≥ n / 2 for each vertex of G.
According to Ore’s theorem deg (v) + deg (w) ≥ n for every n and v not connected by an edge is sufficient condition for a graph to be hamiltonian.
If |E(G)| ≥ 1 / 2 * [(n – 1) (n – 2)] then graph is connected but it doesn’t guaranteed to be Hamiltonian Graph.
(a) and (c) is correct regarding to Hamiltonian Graph.
So, option (C) is correct.

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