Vertex-Transitive $q$-Complementary Uniform Hypergraphs

Abstract

For a positive integer $q$, a $k$-uniform hypergraph $X=(V,E)$ is $q$-complementary if there exists a permutation $\theta$ on $V$ such that the sets $E, E^{\theta}, E^{\theta^2},\ldots, E^{\theta^{q-1}}$ partition the set of $k$-subsets of $V$. The permutation $\theta$ is called a $q$-antimorphism of $X$. The well studied self-complementary uniform hypergraphs are 2-complementary. For an integer $n$ and a prime $p$, let $n_{(p)}=\max\{i:p^i \text{divides} n\}$. In this paper, we prove that a vertex-transitive $q$-complementary $k$-hypergraph of order $n$ exists if and only if $n^{n_{(p)}}\equiv 1 (\bmod q^{\ell+1})$ for every prime number $p$, in the case where $q$ is prime, $k = bq^\ell$ or $k=bq^{\ell}+1$ for a positive integer $b < k$, and $n\equiv 1(\bmod q^{\ell+1})$. We also find necessary conditions on the order of these structures when they are $t$-fold-transitive and $n\equiv t (\bmod q^{\ell+1})$, for $1\leq t < k$, in which case they correspond to large sets of isomorphic $t$-designs. Finally, we use group theoretic results due to Burnside and Zassenhaus to determine the complete group of automorphisms and $q$-antimorphisms of these hypergraphs in the case where they have prime order, and then use this information to write an algorithm to generate all of these objects. This work extends previous, analagous results for vertex-transitive self-complementary uniform hypergraphs due to Muzychuk, Potočnik, Šajna, and the author. These results also extend the previous work of Li and Praeger on decomposing the orbitals of a transitive permutation group.

The core of a vertex transitive complementary prism of a lexicographic product

“…In [22] it was shown that Γ Γ is a core whenever Γ is strongly regular and self-complementary (see also the arXiv version [26,Theorem 5.7]). The same conclusion was obtained in [23] (see also [26, Theorem 5.10]) for all vertex-transitive self-complementary graphs Γ, provided that Γ is corecomplete, i.e. Γ is either a core or it has an endomorphism that maps Γ onto a maximum clique.…”

“…In [26,Corollary 3.8] it was recently proved that the complementary prism Γ Γ is vertextransitive if and only if Γ is vertex-transitive and self-complementary. In [23] it was proved that a vertex-transitive complementary prism Γ Γ is a core whenever Γ is a core or its core is complete. In Corollaries 4.3 and 4.7 we consider the only vertex-transitive selfcomplementary graphs the author is aware of, which are neither cores nor their cores are complete.…”

“…It is obvious that each regular self-complementary graph on n vertices is ( n−1 2 )-regular. Consequently, the hand-shaking lemma implies that n = 4m + 1 for some nonnegative integer m, and Lemma 2.4 follows from the results in [27] or [30] (see also [7, p. 12] and [23]).…”

“…Γ is either a core or it has an endomorphism that maps Γ onto a maximum clique. In general, the existence of a vertex-transitive self-complementary graph Γ, such that Γ Γ is not a core, was stated as an open problem ( [23], [26,Open Problem 5.12]).…”

“…They are in the form of a lexicographic product Γ = Γ 1 [Γ 2 ] for specially chosen graphs Γ 1 and Γ 2 . It turns out that also in these cases the complementary prism Γ Γ is a core (see Corollaries 4.3 and 4.7), which means that the problem stated in [23] remains open.…”