October 5, 2023
October 5, 2023
October 5, 2023
###### GATE 2023
October 5, 2023
 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) only B (b) and (c) C (c) only D (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.
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.