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On the spectrum of $r$-orthogonal Latin squares of different orders | ||
| Transactions on Combinatorics | ||
| مقاله 52، دوره 5، شماره 2، شهریور 2016، صفحه 41-51 اصل مقاله (238.95 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22108/toc.2016.11665 | ||
| نویسندگان | ||
| Hanieh Amjadi1؛ Nasrin Soltankhah* 1؛ Naji Shajarisales2؛ Mehrdad Tahvilian3 | ||
| 1Alzahra University | ||
| 2Max Planck Institute for Intelligent Systems | ||
| 3Sharif University of Technology | ||
| چکیده | ||
| Two Latin squares of order $n$ are orthogonal if in their superposition, each of the $n^{2}$ ordered pairs of symbols occurs exactly once. Colbourn, Zhang and Zhu, in a series of papers, determined the integers $r$ for which there exist a pair of Latin squares of order $n$ having exactly $r$ different ordered pairs in their superposition. Dukes and Howell defined the same problem for Latin squares of different orders $n$ and $n+k$. They obtained a non-trivial lower bound for $r$ and solved the problem for $k \geq \frac{2n}{3} $. Here for $k < \frac{2n}{3}$, some constructions are shown to realize many values of $r$ and for small cases $(3\leq n \leq 6)$, the problem has been solved. | ||
| کلیدواژهها | ||
| Latin square؛ Orthogonal Latin square؛ $r$-Orthogonal Latin square؛ $r$-Orthogonality spectrum؛ Transversal | ||
| مراجع | ||
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[1] G. B. Belyavskaya, $r$-Orthogonal quasigroups I, Math. Issled., 39 (1976) 32–39. [2] G. B. Belyavskaya, $r$-Orthogonal quasigroups II, Math. Issled., 43 (1976) 39–49. [3] G. B. Belyavskaya, $r$-Orthogonal Latin squares, in: J. Dénes and A. D. Keedwell (Editors), Latin Squares: New Developments, Elsevier, North-Holland, Amsterdam, 1992. [4] C. J. Colbourn and L. Zhu, The spectrum of r-orthogonal Latin squares, C. J. Colbourn, E. S. Mahmoodian (Editors), Combinatorics advances (Tehran, 1994), Kluwer Academic Press, Dordrecht, 1995 49–75. [5] P. Dukes and J. Howell, The orthogonality spectrum for Latin squares of different orders, Graphs Combin., 29 (2013) 71–78. [6] J. Howell, The intersection problem and different pairs problem for Latin squares, Ph. D. dissertation, University of Victoria, 2010. [7] H. J. Ryser, A combinatorial theorem with an application to Latin rectangles, Proc. Amer. Math. Soc., 2 (1951) 550–552. [8] L. Zhu and H. Zhang, A few more $r$-orthogonal Latin squares, Discrete Math., 238 (2001) 183–191. [9] L. Zhu and H. Zhang, Completing the spectrum of $r$-orthogonal Latin squares, Discrete Math., 268 (2003) 343–349. | ||
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