GATE | GATE-CS-2015 (Set 2) | Question 42

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

Answer: (D)

Explanation: An n-vertex self-complementary graph has exactly half number of edges of the complete graph, i.e., n(n − 1)/4 edges, and (if there is more than one vertex) it must have diameter either 2 or 3. Since n(n −1) must be divisible by 4, n must be congruent to 0 or 1 mod 4; for instance, a 6-vertex graph cannot be self-complementary.


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