UGC-NET | UGC NET CS 2015 Dec – II | Question 2

Which of the following statement(s) is/are false ?
(a) A connected multigraph has an Euler Circuit if and only if each of its vertices has even degree.
(b) A connected multigraph has an Euler Path but not an Euler Circuit if and only if it has exactly two vertices of odd degree.
(c) A complete graph (K_n) has a Hamilton Circuit whenever n ≥ 3.
(d)A cycle over six vertices (C₆) is not a bipartite graph but a complete graph over 3 vertices is bipartite.
Codes:

(A) (a) only
(B) (b) and (c)
(C) (c) only
(D) (d) only

Answer: (D)
Explanation:

• A connected multigraph has an Euler Circuit if and only if each of its vertices has even degree.Correct
• A connected multigraph has an Euler Path but not an Euler Circuit if and only if it has exactly two vertices of odd degree.Correct
• A complete graph (K_n) has a Hamilton Circuit whenever n ≥ 3.Correct
• A cycle over six vertices (C₆) is not a bipartite graph but a complete graph over 3 vertices is bipartite.Incorrect

So, option (D) is cocrrect.

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