Construction of eleven and thirteen mutually orthogonal path-squares | ||
| Menoufia Journal of Electronic Engineering Research | ||
| Article 5, Volume 18, Issue 1, January 2008, Pages 61-72 | ||
| Document Type: Original Article | ||
| DOI: 10.21608/mjeer.2008.65216 | ||
| Author | ||
| R. A. El-Shanawany | ||
| Dept. of Physics and Engineering Mathematics, Faculty of Electronic Engineering, Menoufya University, EGYPT | ||
| Abstract | ||
| In this paper, we give an introduction to an area of combinatorial theory which deals with. mutually orthogonal (MOGSs) where is a spanning subgraph of That is, we will investigate the maximal number of a collection of of order n which are orthogonal in pairs. The most general object studied in this theory is a set of the so-called mutually orthogonal Latin squares (MOLSs) of order which are equivalent to mutually orthogonal if . That is, this approach is an extension of MOLSs. Our objective in this paper is to discuss two direct constructions from conjecture in [7] for all prime numbers to obtain MOGSs where . | ||
| Keywords | ||
| aph decompositions; MOLSs; Orthogonal double cover | ||
| References | ||
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[1] B. Ganter, H.-D.O.F. Gronau, R. C. Mullin, ”On Orthogonal Double Covers Of ”, Ars Combin., Vol. 37, pp. 209-221, 1994.
[2] B. Ganter, H.-D.O.F. Gronau, ”On Two Conjectures Of Demetrovics, Füredi and Katona On Partitions”, Discrete Math., Vol. 88, pp. 155-194. 1991.
[3] F. E. Bennet, L. Wu, ”On Minimum Matrix Representation Of Closure Operations”, Discrete Appl. Math., Vol. 26, pp. 25-40, 1990.
[4] H.-D.O.F. Gronau, S. Hartman, M. Grüttmüller, U. Leck, V. Leck, ”On Orthogonal Double Covers Of Graphs”, Des. Codes Cryptogr., Vol. 27, pp. 49-91, 2002.
[5] H.-D.O.F. Gronau, R.C. Mullin, A. Rosa, ”Orthogonal Double Covers Of Complete Graphs By Trees”, Graphs Combin., Vol. 13, pp. 251-262, 1997.
[6] R. El-Shanawany, H.-D.O.F. Gronau, and Martin Grüttmüller. ”Orthogonal Double Covers Of By Small Graphs”, Discrete Appl. Math., Vol. 138, pp. 47-63, 2004.
[7] R. El-Shanawany, ”Orthogonal Double Covers Of Complete Bipartite Graphs”, Ph.D. Thesis, Universität Rostock, 2002.
[8] V. Leck, ”Orthogonal Double Covers Of ”, Ph.D. Thesis, Universtät Rostock, 2000.
[9] B. Alspach, K. Heinrich, and G. Liu. Orthgonal factorizations of graphs. In J.H. Dinitz and D.R. Stinson, editors, Contermporary design theory, chapter 2. Wiley,New York, 1992.
[10] D. Betten. Zum Satz von Euler-Terry. Math. Naturewiss. Unterr., 36:449-453, 1983.
[11] T. Beth, D. Jungnikel, and H. Lenz. Design Theory. CombridgeUniversity Press, Cambridge, 1986..
[12] [12] R.C. Bose and S.S. Shrikhande. on the falsity of euler's about the non-existence of two orthogonal latin squares of order 4t+2. Proc. Nat. Acad. Sci., 45:734-737, 1959.
[13] R.C. Bose, S.S. Shrikhande, and E.T. Parker. Further results on the construction of mutually orthogonal latin squares and the falsity of euler's conjucture. Canda. J. Math., 12:189-203, 1960.
[14] C.J. Colbourn and J.H. Dinitz, editors. The CRC Handbook of Combinatorial Designs. CRC Press, Boca Raton, FL, 1996.
[15] L. Euler. Recherches sur une nouveau especce de quarres magiques. In Leonhardi Euleri opera omina. Ser. Prima, 7:291-392, 1900.
[16] C.W.H. Lam, Thiel, and S.Swiercs. The non-existence of projective planes of order 10. Canad. J.Math., 41:1117-1123, 1989.
[17] R.Mendelsohn, and A.Rosa. one-factorizations of the complete graph. A survey J. Graph theory, 9:43-65, 1985.
[18] E.T.Parker. Construction of some sets of mutually orthogonal latin squares. Proc. Amer. Math. Soc., 10:946-949, 1959.
[19] E.T.Parker. orthogonal latin squares. Proc. Nat. Acad. Sci. U.S.A., 45:859-862, 1959.
[20] D.R. Stinson. A short proof of the nonexistence of a pair of orthogonal latin squares of order six. J.Combin. Theory A, 36:373-376, 1984.
[21] M.G. Tarry. Le probleme des 36 officiers. C.R. Assoc. France Av. Sci., 29:170-203, 1900.
[22] W.D. Wallis. one-factorizations of the complete graph.
[23] W.D. Wallis. Combinatorial Designs. Marcel, New York, 1988. | ||
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