It can be easily shown that if a graph is self-complementary and regular then the number of vertices, $n$, is equal to $4k +1$ for some $k \in \mathbb{Z}$.

But, how to we prove (prove by constructing) that there is a self-complementary regular graph for $n = 4k +1$


Here is a source for a construction:

S.B. Rao, On regular and strongly-regular self-complementary graphs. Discrete Mathematics 54 (1985), pp. 73–82.

See Theorem 2.3. This solution seems to answer a more specific question, so it's likely that there is a simpler answer to your question, which I would be interested in.


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