A class of self‐complementary graphs and lower bounds of some ramsey numbers

Clapham, C. R. J.

doi:10.1002/jgt.3190030311

Cited by 20 publications

(13 citation statements)

References 3 publications

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“…, and let L i < Aut(R i ) = GL(2, p i ) be isomorphic to SL (2,5). Let σ i ∈ Z(Aut(R i )) have order divisible by 8.…”

Section: Construction 32mentioning

confidence: 99%

“…We may assume σ ∈ X v . As SL (2,5) has no subgroup of index 2, σ / ∈ SL(2, 5). Since σ normalises G v , σ normalises SL(2, 5), and we may choose σ ∈ z .…”

Section: Lemma 43 Assume That X Is Primitive and Insoluble Then We mentioning

confidence: 99%

“…In the ideal case, Γ is vertex-transitive. Because of this, self-complementary vertex-transitive graphs have been effectively used as models to find good lower bounds of Ramsey numbers (see [4,5,9,23] for references).…”

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confidence: 99%

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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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“…, and let L i < Aut(R i ) = GL(2, p i ) be isomorphic to SL (2,5). Let σ i ∈ Z(Aut(R i )) have order divisible by 8.…”

Section: Construction 32mentioning

confidence: 99%

“…We may assume σ ∈ X v . As SL (2,5) has no subgroup of index 2, σ / ∈ SL(2, 5). Since σ normalises G v , σ normalises SL(2, 5), and we may choose σ ∈ z .…”

Section: Lemma 43 Assume That X Is Primitive and Insoluble Then We mentioning

confidence: 99%

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On finite self-complementary metacirculants

Rao

Song

2014

J Algebr Comb

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“…Self-complementary vertex-transitive graphs have been used as models for finding lower bounds of Ramsey This paper was partially supported by the National Natural Science Foundation of China (11301484, 11231008 and 11461077). c 2015 Australian Mathematical Publishing Association Inc. 0004-9727/2015 $16.00 [2] Self-complementary vertex-transitive graphs 239 numbers (see [4,5,10,19]). More recently, the study of self-complementary vertextransitive graphs has been significantly developed by Li and Praeger [12] and their joint work with Guralnick and Saxl [11] for the vertex-primitive case.…”

Section: Introductionmentioning

confidence: 99%

Automorphism Groups of Self-Complementary Vertex-Transitive Graphs

Huang

Pan

Ding

et al. 2015

Bull. Aust. Math. Soc.

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Li et al. ['On finite self-complementary metacirculants ', J. Algebraic Combin. 40 (2014), 1135-1144 proved that the automorphism group of a self-complementary metacirculant is either soluble or has A 5 as the only insoluble composition factor, and gave a construction of such graphs with insoluble automorphism groups (which are the first examples of self-complementary graphs with this property). In this paper, we will prove that each simple group is a subgroup (so is a section) of the automorphism groups of infinitely many self-complementary vertex-transitive graphs. The proof involves a construction of such graphs. We will also determine all simple sections of the automorphism groups of self-complementary vertex-transitive graphs of 4-power-free order.2010 Mathematics subject classification: primary 20B15; secondary 20B30, 05C25.

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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].…”

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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A class of self‐complementary graphs and lower bounds of some ramsey numbers

Cited by 20 publications

References 3 publications

On finite self-complementary metacirculants

On finite self-complementary metacirculants

Automorphism Groups of Self-Complementary Vertex-Transitive Graphs

On partitioning the orbitals of a transitive permutation group

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