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October 5, 2023GATE 2023
October 5, 2023UGC NET CS 2015 Dec paper2
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.
(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.
(a) only


(b) and (c)


(c) only


(d) only

Question 2 Explanation:
(a)TRUE: A connected multigraph has an Euler Circuit if and only if each of its vertices has even degree.
(b)TRUE: 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)TRUE: A complete graph (K _{n} ) has a Hamilton Circuit whenever n ≥ 3.
(d) FALSE: A cycle over six vertices (C_{ 6} ) is not a bipartite graph but a complete graph over 3 vertices is bipartite.
(b)TRUE: 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)TRUE: A complete graph (K _{n} ) has a Hamilton Circuit whenever n ≥ 3.
(d) FALSE: A cycle over six vertices (C_{ 6} ) is not a bipartite graph but a complete graph over 3 vertices is bipartite.
Correct Answer: D
Question 2 Explanation:
(a)TRUE: A connected multigraph has an Euler Circuit if and only if each of its vertices has even degree.
(b)TRUE: 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)TRUE: A complete graph (K _{n} ) has a Hamilton Circuit whenever n ≥ 3.
(d) FALSE: A cycle over six vertices (C_{ 6} ) is not a bipartite graph but a complete graph over 3 vertices is bipartite.
(b)TRUE: 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)TRUE: A complete graph (K _{n} ) has a Hamilton Circuit whenever n ≥ 3.
(d) FALSE: A cycle over six vertices (C_{ 6} ) is not a bipartite graph but a complete graph over 3 vertices is bipartite.
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