On regular and strongly-regular self-complementary graphs

Rao, S. B.

doi:10.1016/0012-365x(85)90063-9

Cited by 36 publications

(25 citation statements)

References 3 publications

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“…Self-complementary circulants have been studied for a long time and have been taken as good models for studying other combinatorial objects, such as Ramsey numbers and communication networks. The first family of self-complementary circulants was constructed by Sachs in 1962, and since then self-complementary circulants have been widely studied, see [6,19,22,24,25,28] for the work till 1980s. In 1990s, the orders of self-complementary circulants were determined, see [1] for Alspach-Morris-Vilfred's proof and [7] for an alternative approach.…”

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confidence: 99%

On finite self-complementary metacirculants

Rao

Song

2014

J Algebr Comb

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We prove that the automorphism group of a self-complementary metacirculant is either soluble or has A 5 as the only insoluble composition factor, extending a result of Li and Praeger which says the automorphism group of a self-complementary circulant is soluble. The proof involves a construction of self-complementary metacirculants which are Cayley graphs and have insoluble automorphism groups. To the best of our knowledge, these are the first examples of self-complementary graphs with this property.

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confidence: 99%

On finite self-complementary metacirculants

Rao

Song

2014

J Algebr Comb

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“…However, it is important to note that these existence results do not imply that condition (1) of Theorem 1.1 is sucient for t ∈ {1, 2}, since there is no guarantee that two designs in the large sets constructed in these papers are isomorphic. To date, the only existence results for regular and 2-subset-regular self-complementary k-hypergraphs are those due to Rao, Poto£nik, ajna, and Knor [10,11,13] mentioned in the last section.…”

Section: Connection To Design Theorymentioning

confidence: 99%

“…The case where k = 2 was handled constructively by Rao [13], but there is also a proof due to Wilson [15]. Poto£nik and ajna handled the case where k = 3 and t = 1 [11], and Knor and Poto£nik handled the case where k = 3 and t = 2 [10].…”

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confidence: 99%

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Constructing regular self-complementary uniform hypergraphs

Gosselin¹

2011

J. Combin. Designs

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In this paper, we examine the possible orders of t-subset-regular selfcomplementary k-uniform hypergraphs, which form examples of large sets of two isomorphic t-designs. We reformulate Khosrovshahi and TayfehRezaie's necessary conditions on the order of these structures in terms of the binary representation of the rank k, and these conditions simplify to a more transparent relation between the order n and rank k in the case where k is a sum of consecutive powers of 2. Moreover, we present new constructions for 1-subset-regular self-complementary uniform hypergraphs, and prove that these necessary conditions are sucient for all k, in the case where t = 1.

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“…Vertex-transitive self-complementary graphs have received considerable attention in the literature, see for example [14,17,18,20,25,27], and they have been used to investigate Ramsey numbers [3,4,5]. Most of the known vertex-transitive selfcomplementary graphs are Cayley graphs, see for example [14,18,25,19,23]; the first infinite family of vertex-transitive self-complementary graphs that are not Cayley graphs was constructed recently in [17]. With regard to TODs of arbitrary index k, we give necessary and sufficient conditions for the existence of a k-TOD in Proposition 3.3, and the proof of this result includes a general construction for them.…”

Section: Introductionmentioning

confidence: 99%

On partitioning the orbitals of a transitive permutation group

Praeger

2002

Trans. Amer. Math. Soc.

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Abstract. Let G be a permutation group on a set Ω with a transitive normal subgroup M . Then G acts on the set Orbl(M, Ω) of nontrivial M -orbitals in the natural way, and here we are interested in the case where Orbl(M, Ω) has a partition P such that G acts transitively on P. The problem of characterising such tuples (M, G, Ω, P), called TODs, arises naturally in permutation group theory, and also occurs in number theory and combinatorics. The case where |P| is a prime-power is important in algebraic number theory in the study of arithmetically exceptional rational polynomials. The case where |P| = 2 exactly corresponds to self-complementary vertex-transitive graphs, while the general case corresponds to a type of isomorphic factorisation of complete graphs, called a homogeneous factorisation. Characterising homogeneous factorisations is an important problem in graph theory with applications to Ramsey theory. This paper develops a framework for the study of TODs, establishes some numerical relations between the parameters involved in TODs, gives some reduction results with respect to the G-actions on Ω and on P, and gives some construction methods for TODs.

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On regular and strongly-regular self-complementary graphs

Cited by 36 publications

References 3 publications

On finite self-complementary metacirculants

On finite self-complementary metacirculants

Constructing regular self-complementary uniform hypergraphs

On partitioning the orbitals of a transitive permutation group

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