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 (C_{6}) 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 (C
_{6}) is not a bipartite graph but a complete graph over 3 vertices is bipartite.**Incorrect**

So, option (D) is cocrrect.

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