Self-dual orientable embeddings of Kn

Pengelley, David

doi:10.1016/0095-8956(75)90063-5

Cited by 17 publications

(10 citation statements)

References 3 publications

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“…By Euler's formula, orientable selfdual embeddings of K n may exist only for n ≡ 0, 1 (mod 4). Pengelley [7] constructed orientable self-dual embeddings of K n for all n ≡ 0, 1 (mod 4), n ≥ 4, but one can check that the embeddings (except the case n = 2 r ) do not have the property that every face is incident with n−1 different vertices of K n .…”

Section: The Resultsmentioning

confidence: 99%

Finite fields and the 1‐chromatic number of orientable surfaces

Korzhik

2009

Journal of Graph Theory

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Section: The Resultsmentioning

confidence: 99%

Finite fields and the 1‐chromatic number of orientable surfaces

Korzhik

2009

Journal of Graph Theory

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“…Pengelly in Ref. [13] released 2 s (t − 1) elements with distinct inverses, and his choose a 1 , a 2 , a 3 ,. .…”

Section: Self-dual Orientable Embeddings Of Complete Graphmentioning

confidence: 99%

“…In 2013 Korzhik [21] studied generating nonisomorphic quadrangular embeddings of a complete graph. In [22] [13], Archdeacon et al [14] and Archdeacon [25] in self-dual orientable embeddings of the complete graphs K 4r+1 and K 4s , (r ≥ 1 and s ≥ 2), complete bipartite graphs and complete multipartite graphs by current graphs and rotation schemes [23]. Binary quantum codes are defined by pair (H X , H Z ) of Z 2 -matrices with H X H T Z = 0.…”

Section: Introductionmentioning

confidence: 99%

Classes of quantum codes derived from self-dual orientable embeddings of complete multipartite graphs

Naghipour

Jafarizadeh

2017

Quantum Inf Process

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This paper presents four new classes of binary quantum codes with minimum distance 3 and 4, namely Class-I, Class-II, Class-III and Class-IV. The classes Class-I and Class-II are constructed based on self-dual orientable embeddings of the complete graphs K 4r+1 and K 4s and by current graphs and rotation schemes. The parameters of two classes of quantum codes are [[2r(4r + 1), 2r(4r − 3), 3]] and [[2s(4s−1), 2(s−1)(4s−1), 3]] respectively, where r ≥ 1 and s ≥ 2. For these quantum codes, the code rate approaches 1 as r and s tend to infinity. The Class-III with rate 1 2 and with minimum distance 4 is constructed by using self-dual embeddings of complete bipartite graphs. The parameters of this class are [[rs, (r−2)(s−2) 2, 4]], where r and s are both divisible by 4. The proposed Class-IV is of minimum distance 3 and code length n = (2r+1)s 2 . This class is constructed based on self-dual embeddings of complete tripartite graph Krs,s,s and its parameters are [[(2r+1)s 2 , (rs−2)(s−1), 3]], where r ≥ 2 and s ≥ 2.

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“…He made use of some elementary finite field theory to construct orientable self-dual embeddings of K, where q is a prime and q = 1 (mod 4). More orientable self-dual embeddings were subsequently produced by Biggs (1971), White (1974), Pengelley (1975) and Bouchet (1975), using a variety of techniques. All of these constructions are special cases of the method introduced by Stahl (1978b).…”

Section: Generative M-valuationsmentioning

confidence: 99%

The embeddings of a graph—A survey

Stahl

1978

Journal of Graph Theory

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Topological graph theory seeks to find answers to the question of how graphs map into surfaces. This paper surveys the information now available about the range of a graph, namely, the set of surfaces on which the graph can be "neatly" embedded. Several other closely related topics, such as irreducible graphs, coloring problems, and crossing numbers, are ignored. As is quite often the case with mathematical theories, this discipline developed in a rather haphazard manner. Many isolated results existed before the practitioners became aware of the fact that they were developing a theory. The turning point occurred in 1968, when Ringel and Youngs completed their proof of the Heawood conjecture. Their proof, in addition to settling an old unsolved problem, also reinforced the significance of the rotation systems. It is the author's belief tnat these rotation systems, together with the generalized embedding schemes can, and should, become the main tool in all investigations concerning the embeddings of a graph. This survey is written from that point of view. After defining the scope of the area surveyed, this paper proceeds to discuss the significance of the rotation systems and embedding schemes. Several theorems of a general nature are listed. Attention is then focused on the maximum and minimum genera of a graph. Discussion of the first of these is deferred to another survey article by R. Ringeisen to appear in a subsequent issue. The various methods developed by researchers in this area for determining the (minimum) genus are then described. This is followed by a listing of all the theoretical information that is available about the genus parameter. The paper includes two tables that exhibit most of the graphs with known genus.The unifying theme of this survey is the following problem:Q. Given a graph G, characterize the closed surfaces on which G can be 2-cell embedded.

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Self-dual orientable embeddings of Kn

Cited by 17 publications

References 3 publications

Finite fields and the 1‐chromatic number of orientable surfaces

Finite fields and the 1‐chromatic number of orientable surfaces

Classes of quantum codes derived from self-dual orientable embeddings of complete multipartite graphs

The embeddings of a graph—A survey

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