Pseudo orthogonal Latin squares
-
Shahab Faruqi
,
S. A. Katre
Abstract
Two Latin squares A, B of order n are called pseudo orthogonal if for any 1 ≤ i, j ≤ n there exists a k, 1 ≤ k ≤ n, such that A(i, k) = B(j, k). We prove that the existence of a family of m mutually pseudo orthogonal Latin squares of order n is equivalent to the existence of a family of m mutually orthogonal Latin squares of order n. We also obtain exact values of clique partition numbers of several classes of complete multipartite graphs and of the tensor product of complete graphs.
Note: Originally published in Diskretnaya Matematika (2020) 32, №3, 113–129 (in Russian).
Acknowledgement
The authors thank Bhaskaracharya Pratishthana (Institute of Mathematics), Pune, for certain facilities and A. Zubkov for his helpful comment regarding computational complexity. The second author thanks support from Lokmanya Tilak Chair, S. P. Pune University, for research facilities.
References
[1] Bolshakova N. S., “The intersection number of complete r-partite graphs”, Discrete Math. Appl., 18:2 (2008), 187–197.10.1515/DMA.2008.015Search in Google Scholar
[2] Bose R.C., Shrikhande S.S., Parker E.T., “Further results on the construction of mutually orthogonal Latin squares and the falsity of Euler’s conjecture”, Canad. J. Math., 12 (1960), 189–203.10.4153/CJM-1960-016-5Search in Google Scholar
[3] Erdös P., Goodman A.W., Pósa L., “The representation of a graph by set intersections”, Canad. J. Math., 18 (1966), 106–112.10.4153/CJM-1966-014-3Search in Google Scholar
[4] Hedayat A., “An application of sum composition: A self orthogonal Latin square of order ten”, J. Comb. Theory, Ser. A, 14:2 (1973), 256–260.10.1016/0097-3165(73)90027-7Search in Google Scholar
[5] Hon W.-K. et al., “Edge-clique covers of the tensor product”, Proc. 10th Int. Conf. Algor. Aspects Inf. Manag., Vancouver,, 2014, 66–74.10.1007/978-3-319-07956-1_7Search in Google Scholar
[6] Katre S.A., Yahyaei L., Arumugam S., “Coprime index of a graph”, Electr. Notes Discr. Math., 60:Suppl. C (2017), 77–82.10.1016/j.endm.2017.06.011Search in Google Scholar
[7] Keedwell A.D., Dénes J., Latin Squares and their Applications, Elsevier, 2015.Search in Google Scholar
[8] Laywine C.F., Mullen G.L., Discrete Mathematics Using Latin Squares, J. Wiley & Sons, 1998.Search in Google Scholar
[9] Lindner C., Rodger C., Design Theory, CRC Press, Boca Raton, FL, 1997.Search in Google Scholar
[10] Rodger C.A., Lindner C.C., Design theory, Chapman and Hall/CRC, 2008.Search in Google Scholar
[11] Stinson D.R., Combinatorial Designs: Constructions and Analysis, Springer Sci.& Busin. Media, 2007.Search in Google Scholar
[12] West D. B., Introduction to Graph Theory, Prentice Hall, India Learning Private Ltd, 2002.Search in Google Scholar
© 2021 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Reduction of the integer factorization complexity upper bound to the complexity of the Diffie–Hellman problem
- Pseudo orthogonal Latin squares
- Panchromatic colorings of random hypergraphs
- On the connectivity of configuration graphs
- Asymptotics with remainder term for moments of the total cycle number of random A-permutation
- On the dependence of the complexity and depth of reversible circuits consisting of NOT, CNOT, and 2-CNOT gates on the number of additional inputs
- Letter to the Editor
Articles in the same Issue
- Frontmatter
- Reduction of the integer factorization complexity upper bound to the complexity of the Diffie–Hellman problem
- Pseudo orthogonal Latin squares
- Panchromatic colorings of random hypergraphs
- On the connectivity of configuration graphs
- Asymptotics with remainder term for moments of the total cycle number of random A-permutation
- On the dependence of the complexity and depth of reversible circuits consisting of NOT, CNOT, and 2-CNOT gates on the number of additional inputs
- Letter to the Editor
