Vertex-bipancyclicity of the generalized honeycomb tori

Shih, Yuan-Kang; Wu, Yi-Chien; Kao, Shin-Shin; Tan, Jinglu

doi:10.1016/j.camwa.2008.07.030

Cited by 3 publications

(3 citation statements)

References 37 publications

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“…Cycles whose lengths are congruent to 2 modulo 4 are straightforward as the following easily proved result indicates. [15]. We summarize their results in Table 1.…”

Section: Cycle Structurementioning

confidence: 99%

“…Using methods from [15], it is not hard to show that there are cycles of all lengths that are multiples of 4 lying between 12 and n inclusive. This means the only possible missing cycle lengths are 4 and 8.…”

Section: Cycle Structurementioning

confidence: 99%

“…Honeycomb toroidal graphs HTG(m, n, ) for m ≥ 3 and n > 4 have a simple 12-cycle lying in three columns so that lengths 4 and 8 become the only possible missing values once it is seen how to increase cycle lengths by 4 at a time. The cycle spectrum problem was settled for HTG(m, n, ), when m ≥ 3, in [15]. We summarize their results in Table 1.…”

Section: Cycle Structurementioning

confidence: 99%

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Honeycomb toroidal graphs are Cayley graphs

Alspach¹,

Dean²

2009

Information Processing Letters

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“…Cycles whose lengths are congruent to 2 modulo 4 are straightforward as the following easily proved result indicates. [15]. We summarize their results in Table 1.…”

Section: Cycle Structurementioning

confidence: 99%

Section: Cycle Structurementioning

confidence: 99%

Section: Cycle Structurementioning

confidence: 99%

See 1 more Smart Citation

Honeycomb toroidal graphs are Cayley graphs

Alspach¹,

Dean²

2009

Information Processing Letters

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Ring embedding in faulty generalized honeycomb torus – GHT(m, n, n/2)

Hsu

Ling

Kao

et al. 2010

International Journal of Computer Mathematics

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The honeycomb torus HT(m) is an attractive architecture for distributed processing applications. For analysing its performance, a symmetric generalized honeycomb torus, GHT(m, n, n/2), with m ≥ 2 and even n ≥ 4, where m + n/2 is even, which is a 3-regular, Hamiltonian bipartite graph, is operated as a platform for combinatorial studies. More specifically, GHT(m, n, n/2) includes GHT(m, 6m, 3m), the isomorphism of the honeycomb torus HT(m). It has been proven that any GHT(m, n, n/2) − e is Hamiltonian for any edge e ∈ E(GHT(m, n, n/2)). Moreover, any GHT(m, n, n/2) − F is Hamiltonian for any F = {u, v} with u ∈ B and v ∈ W , where B and W are the bipartition of V (GHT(m, n, n/2)) if and only if n ≥ 6 or m = 2, n ≥ 4.

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On Three Constructions of Nanotori

et al. 2020

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There are three different approaches for constructing nanotori in the literature: one with three parameters suggested by Altshuler, another with four parameters used mostly in chemistry and physics after the discovery of fullerene molecules, and one with three parameters used in interconnecting networks of computer science known under the name generalized honeycomb tori. Altshuler showed that his method gives all non-isomorphic nanotori, but this was not known for the other two constructions. Here, we show that these three approaches are equivalent and give explicit formulas that convert parameters of one construction into the parameters of the other two constructions. As a consequence, we obtain that the other two approaches also construct all nanotori. The four parameters construction is mainly used in chemistry and physics to describe carbon nanotori molecules. Some properties of the nanotori can be predicted from these four parameters. We characterize when two different quadruples define isomorphic nanotori. Even more, we give an explicit form of all isomorphic nanotori (defined with four parameters). As a consequence, infinitely many 4-tuples correspond to each nanotorus; this is due to redundancy of having an “extra” parameter, which is not a case with the other two constructions. This result significantly narrows the realm of search of the molecule with desired properties. The equivalence of these three constructions can be used for evaluating different graph measures as topological indices, etc.

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Vertex-bipancyclicity of the generalized honeycomb tori

Cited by 3 publications

References 37 publications

Honeycomb toroidal graphs are Cayley graphs

Honeycomb toroidal graphs are Cayley graphs

Ring embedding in faulty generalized honeycomb torus – GHT(m, n, n/2)

On Three Constructions of Nanotori

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