On Sequentially Perfect and Sequentially Uniform One Factorizations of Product of Cycles

Jamuna Loganathan; Hemalatha Palanisamy

doi:doi:10.11648/j.dmath.20250701.12

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On Sequentially Perfect and Sequentially Uniform One Factorizations of Product of Cycles

Jamuna Loganathan^*, Hemalatha Palanisamy

Published in International Journal of Discrete Mathematics (Volume 7, Issue 1)

Received: 12 August 2025 Accepted: 28 August 2025 Published: 25 September 2025

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Abstract

Let G be an n-regular graph. A sequentially perfect one-factorization is a decomposition of G into one-factors, such that union of any two consecutive one-factors gives a hamiltonian cycle, and also if the union of two consecutive one-factors forms a two regular graph, then this property is termed as sequentially uniform. For any two graphs G and H their product [Cartesian, Tensor, Wreath] has the vertex set {(g, h); g belongs to V(G) and h belongs to V(H)}. The Cartesian Product G Cartesian Product H has the edge set such that two vertices (g₁, h₁) and (g₂, h₂) are adjacent if they differ in exactly one coordinate, and the corresponding vertices in that graph are adjacent; in the tensor product G Tensor Product H two vertices (g₁, h₁) and (g₂, h₂) are adjacent only if both g₁ is adjacent to g₂ in G and h₁ is adjacent to h₂ in H; in the wreath product G Wreath Product H the edge set is E(G Wreath Product H) = {(g₁, h₁) (g₂, h₂) : g₁g₂ belongs to E(G) or g₁=g₂ and h₁h₂ belongs to E(H)}. In this paper it is proved that for all even m > 2 and odd n ≥ 3, there exists sequentially perfect and sequentially uniform one-factorizations in C_m Cartesian Product C_n, C_m Tensor Product C_n and C_m Wreath Product C_n.

Published in	International Journal of Discrete Mathematics (Volume 7, Issue 1)
DOI	10.11648/j.dmath.20250701.12
Page(s)	14-20
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2025. Published by Science Publishing Group

Keywords

Cycles, Cartesian Product of Graphs, Tensor Product of Graphs, Wreath Product of Graphs, One-factorization, Sequentially Perfect One-factorization, Sequentially Uniform One-factorization

References

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[2]	Bryant D. E., Maenhaut B. M., and Wanless I. M. A family of perfect factorisations of complete bipartite graphs. Journal of Combinatorial Theory, Series A, 98(2): 328-342, 2002. https://doi.org/10.1006/jcta.2001.3240
[3]	Pike D. A. A perfect one-factorization of K₅₆. Journal of Combinatorial Designs, 27(3): 190-196, 2019. https://doi.org/10.1002/jcd.21653
[4]	Dinitz J. H. and Garnick D. K. There are 23 nonisomorphic perfect one-factorizations of K₁₄. Journal of Combinatorial Designs, 4(1): 1-3, 1996. https://doi.org/10.1002/(SICI)1520-6610(1996)4:1<1::AID-JCD1>3.0.CO;2-J
[5]	Dinitz J. H., Dukes P., and Stinson D. R. Sequentially perfect and uniform one-factorizations of the complete graph. The Electronic Journal of Combinatorics, 12(1): R1, 23-30, 2005. https://doi.org/10.37236/1898
[6]	Ganesamurthy S. and Manikandan R. Decomposition of some classes of regular graphs and digraphs into cycles of length 4p. Australasian Journal of Combinatorics, 79(2): 215-233, 2021.
[7]	Gill N. and Wanless I. M. Perfect 1-factorizations of K₁₆. Bulletin of the Australian Mathematical Society, 101(2): 177-185, 2020. https://doi.org/10.1017/S0004972719000856
[8]	Gray C., Zhang P., and Linda L. Graphs and Digraphs. CRC Press, Taylor and Francis Group, 5th edition, 2011. https://doi.org/10.1201/b19731
[9]	Hemalatha P. and Jamuna L. On k-semi-perfect one-factorizations of Cartesian product of graphs. International Journal of Communication Networks and Information Security, 16(1): 6698, 2024. https://doi.org/10.1201/b14892
[10]	Petersen J. The theory of regular graphs. Acta Mathematica, 15(1): 193-220, 1891. https://doi.org/10.1007/BF02392606
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[13]	Mendelsohn E. and Rosa A. One-factorizations of the complete graph: A survey. Journal of Graph Theory, 9(1): 43-65, 1985. https://doi.org/10.1002/jgt.3190090104
[14]	Pisanski T., Shawe-Taylor J., and Mohar B. 1-factorization of the composition of regular graphs. Publications de l’Institut Mathématique (Belgrade), 33(47): 193-198, 1983.
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Loganathan, J., Palanisamy, H. (2025). On Sequentially Perfect and Sequentially Uniform One Factorizations of Product of Cycles. International Journal of Discrete Mathematics, 7(1), 14-20. https://doi.org/10.11648/j.dmath.20250701.12

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ACS Style

Loganathan, J.; Palanisamy, H. On Sequentially Perfect and Sequentially Uniform One Factorizations of Product of Cycles. Int. J. Discrete Math. 2025, 7(1), 14-20. doi: 10.11648/j.dmath.20250701.12

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AMA Style

Loganathan J, Palanisamy H. On Sequentially Perfect and Sequentially Uniform One Factorizations of Product of Cycles. Int J Discrete Math. 2025;7(1):14-20. doi: 10.11648/j.dmath.20250701.12

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@article{10.11648/j.dmath.20250701.12,
  author = {Jamuna Loganathan and Hemalatha Palanisamy},
  title = {On Sequentially Perfect and Sequentially Uniform One Factorizations of Product of Cycles},
  journal = {International Journal of Discrete Mathematics},
  volume = {7},
  number = {1},
  pages = {14-20},
  doi = {10.11648/j.dmath.20250701.12},
  url = {https://doi.org/10.11648/j.dmath.20250701.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.dmath.20250701.12},
  abstract = {Let G be an n-regular graph. A sequentially perfect one-factorization is a decomposition of G into one-factors, such that union of any two consecutive one-factors gives a hamiltonian cycle, and also if the union of two consecutive one-factors forms a two regular graph, then this property is termed as sequentially uniform. For any two graphs G and H their product [Cartesian, Tensor, Wreath] has the vertex set {(g, h); g belongs to V(G) and h belongs to V(H)}. The Cartesian Product G Cartesian Product H has the edge set such that two vertices (g1, h1) and (g2, h2) are adjacent if they differ in exactly one coordinate, and the corresponding vertices in that graph are adjacent; in the tensor product G Tensor Product H two vertices (g1, h1) and (g2, h2) are adjacent only if both g1 is adjacent to g2 in G and h1 is adjacent to h2 in H; in the wreath product G Wreath Product H the edge set is E(G Wreath Product H) = {(g1, h1) (g2, h2) : g1g2 belongs to E(G) or g1=g2 and h1h2 belongs to E(H)}. In this paper it is proved that for all even m > 2 and odd n ≥ 3, there exists sequentially perfect and sequentially uniform one-factorizations in Cm Cartesian Product Cn, Cm Tensor Product Cn and Cm Wreath Product Cn.},
 year = {2025}
}

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TY  - JOUR
T1  - On Sequentially Perfect and Sequentially Uniform One Factorizations of Product of Cycles
AU  - Jamuna Loganathan
AU  - Hemalatha Palanisamy
Y1  - 2025/09/25
PY  - 2025
N1  - https://doi.org/10.11648/j.dmath.20250701.12
DO  - 10.11648/j.dmath.20250701.12
T2  - International Journal of Discrete Mathematics
JF  - International Journal of Discrete Mathematics
JO  - International Journal of Discrete Mathematics
SP  - 14
EP  - 20
PB  - Science Publishing Group
SN  - 2578-9252
UR  - https://doi.org/10.11648/j.dmath.20250701.12
AB  - Let G be an n-regular graph. A sequentially perfect one-factorization is a decomposition of G into one-factors, such that union of any two consecutive one-factors gives a hamiltonian cycle, and also if the union of two consecutive one-factors forms a two regular graph, then this property is termed as sequentially uniform. For any two graphs G and H their product [Cartesian, Tensor, Wreath] has the vertex set {(g, h); g belongs to V(G) and h belongs to V(H)}. The Cartesian Product G Cartesian Product H has the edge set such that two vertices (g1, h1) and (g2, h2) are adjacent if they differ in exactly one coordinate, and the corresponding vertices in that graph are adjacent; in the tensor product G Tensor Product H two vertices (g1, h1) and (g2, h2) are adjacent only if both g1 is adjacent to g2 in G and h1 is adjacent to h2 in H; in the wreath product G Wreath Product H the edge set is E(G Wreath Product H) = {(g1, h1) (g2, h2) : g1g2 belongs to E(G) or g1=g2 and h1h2 belongs to E(H)}. In this paper it is proved that for all even m > 2 and odd n ≥ 3, there exists sequentially perfect and sequentially uniform one-factorizations in Cm Cartesian Product Cn, Cm Tensor Product Cn and Cm Wreath Product Cn.
VL  - 7
IS  - 1
ER  -

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Jamuna Loganathan

Department of Mathematics, Vellalar College for Women, Erode, India

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Hemalatha Palanisamy

Department of Mathematics, Vellalar College for Women, Erode, India

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APA Style

Loganathan, J., Palanisamy, H. (2025). On Sequentially Perfect and Sequentially Uniform One Factorizations of Product of Cycles. International Journal of Discrete Mathematics, 7(1), 14-20. https://doi.org/10.11648/j.dmath.20250701.12

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ACS Style

Loganathan, J.; Palanisamy, H. On Sequentially Perfect and Sequentially Uniform One Factorizations of Product of Cycles. Int. J. Discrete Math. 2025, 7(1), 14-20. doi: 10.11648/j.dmath.20250701.12

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AMA Style

Loganathan J, Palanisamy H. On Sequentially Perfect and Sequentially Uniform One Factorizations of Product of Cycles. Int J Discrete Math. 2025;7(1):14-20. doi: 10.11648/j.dmath.20250701.12

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@article{10.11648/j.dmath.20250701.12,
  author = {Jamuna Loganathan and Hemalatha Palanisamy},
  title = {On Sequentially Perfect and Sequentially Uniform One Factorizations of Product of Cycles},
  journal = {International Journal of Discrete Mathematics},
  volume = {7},
  number = {1},
  pages = {14-20},
  doi = {10.11648/j.dmath.20250701.12},
  url = {https://doi.org/10.11648/j.dmath.20250701.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.dmath.20250701.12},
  abstract = {Let G be an n-regular graph. A sequentially perfect one-factorization is a decomposition of G into one-factors, such that union of any two consecutive one-factors gives a hamiltonian cycle, and also if the union of two consecutive one-factors forms a two regular graph, then this property is termed as sequentially uniform. For any two graphs G and H their product [Cartesian, Tensor, Wreath] has the vertex set {(g, h); g belongs to V(G) and h belongs to V(H)}. The Cartesian Product G Cartesian Product H has the edge set such that two vertices (g1, h1) and (g2, h2) are adjacent if they differ in exactly one coordinate, and the corresponding vertices in that graph are adjacent; in the tensor product G Tensor Product H two vertices (g1, h1) and (g2, h2) are adjacent only if both g1 is adjacent to g2 in G and h1 is adjacent to h2 in H; in the wreath product G Wreath Product H the edge set is E(G Wreath Product H) = {(g1, h1) (g2, h2) : g1g2 belongs to E(G) or g1=g2 and h1h2 belongs to E(H)}. In this paper it is proved that for all even m > 2 and odd n ≥ 3, there exists sequentially perfect and sequentially uniform one-factorizations in Cm Cartesian Product Cn, Cm Tensor Product Cn and Cm Wreath Product Cn.},
 year = {2025}
}

Copy | Download

TY  - JOUR
T1  - On Sequentially Perfect and Sequentially Uniform One Factorizations of Product of Cycles
AU  - Jamuna Loganathan
AU  - Hemalatha Palanisamy
Y1  - 2025/09/25
PY  - 2025
N1  - https://doi.org/10.11648/j.dmath.20250701.12
DO  - 10.11648/j.dmath.20250701.12
T2  - International Journal of Discrete Mathematics
JF  - International Journal of Discrete Mathematics
JO  - International Journal of Discrete Mathematics
SP  - 14
EP  - 20
PB  - Science Publishing Group
SN  - 2578-9252
UR  - https://doi.org/10.11648/j.dmath.20250701.12
AB  - Let G be an n-regular graph. A sequentially perfect one-factorization is a decomposition of G into one-factors, such that union of any two consecutive one-factors gives a hamiltonian cycle, and also if the union of two consecutive one-factors forms a two regular graph, then this property is termed as sequentially uniform. For any two graphs G and H their product [Cartesian, Tensor, Wreath] has the vertex set {(g, h); g belongs to V(G) and h belongs to V(H)}. The Cartesian Product G Cartesian Product H has the edge set such that two vertices (g1, h1) and (g2, h2) are adjacent if they differ in exactly one coordinate, and the corresponding vertices in that graph are adjacent; in the tensor product G Tensor Product H two vertices (g1, h1) and (g2, h2) are adjacent only if both g1 is adjacent to g2 in G and h1 is adjacent to h2 in H; in the wreath product G Wreath Product H the edge set is E(G Wreath Product H) = {(g1, h1) (g2, h2) : g1g2 belongs to E(G) or g1=g2 and h1h2 belongs to E(H)}. In this paper it is proved that for all even m > 2 and odd n ≥ 3, there exists sequentially perfect and sequentially uniform one-factorizations in Cm Cartesian Product Cn, Cm Tensor Product Cn and Cm Wreath Product Cn.
VL  - 7
IS  - 1
ER  -

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