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

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UGC NET Computer Science (Paper 2) Dec 2018 Paper

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Answer 1

Hamiltonian graph:

A Hamiltonian graph is one which contain a Hamiltonian cycle. A Hamiltonian cycle is a cycle in which each vertex is visited exactly once.

Properties of Hamiltonian graph:

1) A graph has a Hamiltonian circuit if each vertex has degree >=3

2) If G= (V, E) has n>=3 vertices and every vertex has degree >=n/2, then G has a Hamilton circuit

3) If G is a graph with n vertices and n>=3, also deg(u) + deg(v) >=n, if u and v are not connected by an edge, then G has a hamiltonion circuit.

4) E(G) = ½(n - 1)(n - 2) + 2