Vertex-transitive self-complementary uniform hypergraphs of prime order

Gosselin, Shonda

doi:10.1016/j.disc.2009.08.011

Cited by 6 publications

(8 citation statements)

References 10 publications

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“…The necessity of condition (1) has been proved previously in the case where q = 2 by Potočnik andŠajna [12], and their proof technique is used in Section 2 in the proof of the necessity of this condition in the general case where q is prime. It has also been shown previously that condition (1) is sufficient in the case where q = 2 [3]. In Section 3, we present a construction for vertex-transitive q-complementary uniform hypergraphs to prove that this condition is also sufficient for every odd prime q, which will complete the proof of Theorem 1.3.…”

Section: Lemma 12 [5]mentioning

confidence: 62%

“…We begin with a construction of vertex-transitive q-complementary uniform hypergraphs of prime power order. These hypergraphs are 'Paley-like' in the sense that the construction uses similar algebraic tools to those used in the construction of the well known Paley graphs in [14], the generalized Paley graphs constructed in [8] and studied in [9], the Peisert graphs in [11], and the Paley uniform hypergraphs in [3,6,12].…”

Section: Constructionsmentioning

confidence: 99%

“…Proof of Theorem 1.3: The necessity of condition (1) follows directly from Theorem 2.3. The sufficiency of condition (1) was proved for the prime q = 2 in [3], so we may assume that q is an odd prime. Since k = bq ℓ or k = bq ℓ + 1 for some positive integer b < q, for any integer m such that 1 < m ≤ k we have ℓ ≥ m (q) .…”

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confidence: 99%

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Vertex-Transitive $q$-Complementary Uniform Hypergraphs

Gosselin

2011

Electron. J. Combin.

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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.

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Section: Lemma 12 [5]mentioning

confidence: 62%

Section: Constructionsmentioning

confidence: 99%

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Vertex-Transitive $q$-Complementary Uniform Hypergraphs

Gosselin

2011

Electron. J. Combin.

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“…This result is of great importance in permutation group theory and has useful applications in combinatorics, where for example it is used to determine the automorphism group of graphs or other combinatorial structures whose size is prime [1,3,10] or the product of two primes [8]. Burnside also proved that the socle of a doublytransitive group is either elementary abelian or nonabelian simple [5,Theorem 4.1B].…”

Section: Introductionmentioning

confidence: 95%

Permutation Groups Containing a Regular Abelian Hall Subgroup

Dobson

Li²,

Spiga

2012

Communications in Algebra

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“…Self-complementary uniform hypergraphs have been extensively studied, see [14,16,[18][19][20]25] and the references therein for self-complementary graphs, and see [7][8][9]23,24] for self-complementary uniform hypergraphs. In particular, Peisert [21] gave a complete classification for symmetric (i.e.…”

Section: Introductionmentioning

confidence: 99%

Edge‐transitive homogeneous factorizations of complete uniform hypergraphs

Chen

2017

Journal of Graph Theory

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For a finite set V and a positive integer k with k≤n:=|V|, letting Vfalse{kfalse} be the set of all k‐subsets of V, the pair scriptKnk:=false(V,Vfalse{kfalse}false) is called the complete k‐hypergraph on V, while each k‐subset of V is called an edge. A factorization of the complete k‐hypergraph Knk of index s≥2, simply a (k,s)‐factorization of order n, is a partition {E1,E2,…,Es} of the edges into s disjoint subsets such that each k‐hypergraph (V,Ei), called a factor, is a spanning subhypergraph of Knk. Such a factorization is homogeneous if there exist two transitive subgroups G and M of the symmetric group of degree n such that G induces a transitive action on the set {E1,E2,…,Es} and M lies in the kernel of this action. In this article, we give a classification of homogeneous factorizations of Knk that admit a group acting transitively on the edges of Knk. It is shown that, for 6≤2k≤n and s≥2, there exists an edge‐transitive homogeneous (k,s)‐factorization of order n if and only if (n,k,s) is one of (32, 3, 5), (32, 3, 31), (33, 4, 5), ()2d,3,false(2d−1false)false(2d−1−1false)3, and (q+1,3,2), where d≥3 and q is a prime power with q≡1(mod4).

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Vertex-transitive self-complementary uniform hypergraphs of prime order

Cited by 6 publications

References 10 publications

Vertex-Transitive $q$-Complementary Uniform Hypergraphs

Vertex-Transitive $q$-Complementary Uniform Hypergraphs

Permutation Groups Containing a Regular Abelian Hall Subgroup

Edge‐transitive homogeneous factorizations of complete uniform hypergraphs

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