Proof: For what positive integers k is it the case that the complement of a k-cycle is also a k-cycle?

Question

I am really confused and would really appreciate help regarding this question.

For what positive integers k is it the case that the complement of a k-cycle is also a k-cycle? Prove your answer.

Answer 1

Hint:

The degree of every node in a cycle is 2, since you are connected to the neighbor before and after you.

The degree of a node in a graph's complement is a simple function of its degree in the original graph and the order of the graph.

Writing these two equations starts your proof.