A note on self-complementary 4-uniform hypergraphs

Abstract

We prove that a permutation $\theta$ is complementing permutation for a $4$-uniform hypergraph if and only if one of the following cases is satisfied: (i) the length of every cycle of $\theta$ is a multiple of $8$, (ii) $\theta$ has $1$, $2$ or $3$ fixed points, and all other cycles have length a multiple of $8$, (iii) $\theta$ has $1$ cycle of length $2$, and all other cycles have length a multiple of $8$, (iv) $\theta$ has $1$ fixed point, $1$ cycle of length $2$, and all other cycles have length a multiple of $8$, (v) $\theta$ has $1$ cycle of length $3$, and all other cycles have length a multiple of $8$. Moreover, we present algorithms for generating every possible $3$ and $4$-uniform self-complementary hypergraphs.