Question

A graph is self-complementary if it is isomorphic to its complement. For all self-complementary graphs on n vertices, \( n \) is _______.

• A. a multiple of 4
• B. even
• C. odd
• D. congruent to 0 mod 4 or 1 mod 4

Hint:

Self-complementary graphs have a specific condition for the number of vertices. This condition restricts \( n \) to values that satisfy \( n \equiv 0 \, (\text{mod } 4) \) or \( n \equiv 1 \, (\text{mod } 4) \).

Answer 1

The Correct Option is D

For a graph to be self-complementary, the number of vertices \( n \) must be such that \( n \equiv 0 \, (\text{mod } 4) \) or \( n \equiv 1 \, (\text{mod } 4) \). This is a necessary condition for the graph and its complement to be isomorphic. 
Therefore, the correct answer is option (4).