UGC-NET | UGC NET CS 2018 Dec – II | Question 38

​If a graph (G) has no loops or parallel edges and if the number of vertices(n) in the graph is n≥3, then the graph G is Hamiltonian if

(i) deg(v) ≥n/3 for each vertex v
(ii) deg(v) + deg(w) ≥ n whenever v and w are not connected by an edge.
(iii) E (G) ≥ 1/3 (n − 1 )(n − 2 ) + 2 

(A) (i) and (iii) only
(B) (ii) and (iii) only
(C) (iii) only
(D) (ii) only

Answer: (D)
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.
  So, statement (i) is false.
• 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.
  So, statement (ii) is True.
  
  
• If |E(G)| ≥ 1 / 2 * [(n – 1) (n – 2)] then graph is connected but it doesn’t guaranteed to be Hamiltonian Graph.
  So, statement (iii) is false.

Option (D) is correct.
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