Cyclotomic graphs and perfect codes

Zhou, Sanming

doi:10.1016/j.jpaa.2018.05.007

Cited by 21 publications

(12 citation statements)

References 40 publications

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“…In the past a few years, perfect codes in Cayley graphs have attracted considerable attention, see, for example, [11,28,29]. In [14], Huang, Xia and Zhou first introduced the concept of a perfect code of a group G. A subset C of G is said to be a perfect code of G if C is a perfect code of some Cayley graph of G. In particular, a subgroup is said to be a subgroup perfect code of G if the subgroup is also a perfect code of G. Also in [14], they gave a necessary and sufficient condition for a normal subgroup of a group G to be a subgroup perfect code of G, and determined all the subgroup perfect codes of dihedral groups and some abelian groups.…”

Section: Introductionmentioning

confidence: 99%

Perfect Codes in Cayley Sum Graphs

Wang

Yang

2022

Electron. J. Combin.

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A subset $C$ of the vertex set of a graph $\Gamma$ is called a perfect code of $\Gamma$ if every vertex of $\Gamma$ is at distance no more than one to exactly one vertex in $C$. Let $A$ be a finite abelian group and $T$ a square-free subset of $A$. The Cayley sum graph of $A$ with respect to the connection set $T$ is a simple graph with $A$ as its vertex set, and two vertices $x$ and $y$ are adjacent whenever $x+y\in T$. A subgroup of $A$ is said to be a subgroup perfect code of $A$ if the subgroup is a perfect code of some Cayley sum graph of $A$. In this paper, we give some necessary and sufficient conditions for a subset of $A$ to be a perfect code of a given Cayley sum graph of $A$. We also characterize all subgroup perfect codes of $A$.

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Section: Introductionmentioning

confidence: 99%

Perfect Codes in Cayley Sum Graphs

Wang

Yang

2022

Electron. J. Combin.

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“…Thus perfect t-codes in Cayley graphs are generalization of perfect t-error-correcting Hamming codes or Lee codes. Perfect codes in Cayley graphs have received considerable attention in recent years; see [10,Section 1] for a brief survey and [3,6,18,20,21] for a few recent papers. In particular, perfect codes in Cayley graphs which are subgroups of the underlying groups are especially interesting since they are generalizations of perfect linear codes [15] in the classical setting.…”

Section: Introductionmentioning

confidence: 99%

“…Ma et al [16] proved that all subgroups of a group are subgroup perfect codes if and only if this group does not contain elements of order 4. Very recently, Zhang and Zhou [20] generalized several results about normal subgroups in [10] to general subgroups, and in particular they proved that every subgroup of a group of odd order or odd index is a subgroup perfect code.…”

Section: Introductionmentioning

confidence: 99%

Perfect codes in vertex-transitive graphs

Wang¹,

Zhang²

2021

Preprint

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Given a graph Γ, a perfect code in Γ is an independent set C of vertices of Γ such that every vertex outside of C is adjacent to a unique vertex in C, and a total perfect code in Γ is a set C of vertices of Γ such that every vertex of Γ is adjacent to a unique vertex in C. To study (total) perfect codes in vertex-transitive graphs, we generalize the concept of subgroup (total) perfect code of a finite group introduced in [10] as follows: Given a finite group G and a subgroup H of G, a subgroup A of G containing H is called a subgroup (total) perfect code of the pair (G, H) if there exists a coset graph Cos(G, H, U ) such that the set consisting of left cosets of H in A is a (total) perfect code in Cos(G, H, U ). We give a necessary and sufficient condition for a subgroup A of G containing H to be a (total) perfect code of the pair (G, H) and generalize a few known results of subgroup (total) perfect codes of groups. We also construct some examples of subgroup perfect codes of the pair (G, H) and propose a few problems for further research.

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“…Perfect codes for groups have been well-studied in the past decade [4,[6][7][8]14]. For instance, Huang, Xia, and Zhou [6] gave a necessary and sufficient condition for a normal subgroup to be a perfect code.…”

Section: Introductionmentioning

confidence: 99%

On non-normal subgroup perfect codes

Behajaina¹,

Roghayeh²,

Sarobidy³

2021

Preprint

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Let X = (V, E) be a graph. A subset C ⊆ V (X) is a perfect code of X if C is a coclique of X with the property that any vertex in V (X) \ C is adjacent to exactly one vertex in C. Given a finite group G with identity element e and H ≤ G, H is a subgroup perfect code of G if there exists an inverse-closed subset S ⊆ G \ {e} such that H is a perfect code of the Cayley graph Cay(G, S) of G with connection set S. In this short note, we give an infinite family of finite groups G admitting a non-normal subgroup perfect code H such that there exists g ∈ G with g 2 ∈ H but (gh) 2 = e, for all h ∈ H; thus, answering a question raised by Wang, Xia, and Zhou in [Perfect sets in Cayley graphs. arXiv preprint arXiv:2006.05100, 2020].

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Cyclotomic graphs and perfect codes

Cited by 21 publications

References 40 publications

Perfect Codes in Cayley Sum Graphs

Perfect Codes in Cayley Sum Graphs

Perfect codes in vertex-transitive graphs

On non-normal subgroup perfect codes

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