Abstract
One-overlapped factorizations of finite groups were introduced by Shinohara (Linear Algebra Appl. 436(4):850–857, 2012) for constructing a new class of thin Lehman matrices. However, all known examples and constructions arise from cyclic or dihedral groups. In this paper, we prove the existence of one-overlapped factorizations of non-abelian groups that are not dihedral and describe a simple construction. The corresponding incidence structures and Lehman matrices are also considered.
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Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system. I. The user language. J. Symb. Comput. 24(3-4), 235–265 (1997). https://doi.org/10.1006/jsco.1996.0125 [Computational algebra and number theory (London, 1993)]
Bridges, W.G., Ryser, H.J.: Combinatorial designs and related systems. J. Algebra 13, 432–446 (1969)
de Caen, D., Gregory, D.A., Hughes, I.G., Kreher, D.L.: Near-factors of finite groups. Ars Comb. 29, 53–63 (1990)
Cornuéjols, G.: Combinatorial optimization: packing and covering. In: CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 74. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2001). https://doi.org/10.1137/1.9780898717105
Lehman, A.: On the width-length inequality and degenerate projective planes. In: Cook, W, Seymour, PD. (eds.) Polyhedral Combinatorics (Morristown, NJ, 1989). DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 1, pp. 101–105. American Mathematical Society, Providence, RI (1990)
Pêcher, A.: Partitionable graphs arising from near-factorizations of finite groups. Discrete Math. 269(1–3), 191–218 (2003). https://doi.org/10.1016/S0012-365X(02)00755-0
Sakuma, T., Shinohara, H.: On circulant thin Lehman matrices. J. Algebraic Comb. 40(4), 939–959 (2014). https://doi.org/10.1007/s10801-014-0514-9
Shinohara, H.: Thin Lehman matrices arising from finite groups. Linear Algebra Appl. 436(4), 850–857 (2012). https://doi.org/10.1016/j.laa.2011.02.047
Wang, J.: A new infinite family of minimally nonideal matrices. J. Comb. Theory Ser. A 118(2), 365–372 (2011). https://doi.org/10.1016/j.jcta.2010.09.004
Acknowledgements
This work was supported by the Alexander von Humboldt Foundation with funds from the German Federal Ministry of Education and Research (BMBF). The author would like to thank Dr. Marco A. Pellegrini for his comments on the previous version of this paper.
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This paper is dedicated to Prof. Spyros S. Magliveras for his 80th birthday.
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Pace, N. One-Overlapped Factorizations of Non-abelian Groups. Graphs and Combinatorics 34, 1459–1467 (2018). https://doi.org/10.1007/s00373-018-1937-1
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DOI: https://doi.org/10.1007/s00373-018-1937-1