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Small embeddings of partial directed cycle systems

Published online by Cambridge University Press:  17 April 2009

C.C. Lindner  and
C.A. Rodger
C.C. Lindner
Affiliation:
Department of Algebra, Combinatorics and Analysis Auburn, University Auburn, AL 36849–5307, United States of America
C.A. Rodger
Affiliation:
Department of Algebra, Combinatorics and Analysis Auburn, University Auburn, AL 36849–5307, United States of America
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Abstract

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In this paper, a generalisation of the Andersen, Hilton, Rodger Theorem for embedding partial idempotent latin squares is proved. This result is then used to prove that a partial directed m-cycle system of order n can be embedded in a directed m-cycle system of order (2n + 1)m if m is odd, of order 2nm if m ≥ 8 is even, 12n + 1 if m = 6 and approximately 2n + √2n if m = 4.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 46 , Issue 2 , October 1992 , pp. 213 - 224
Copyright
Copyright © Australian Mathematical Society 1992

References

[1]Andersen, L.D., Hilton, A.J.W. and Rodger, C.A., ‘A solution to the embedding problem for partial idempotent latin squares’, J. London Math. Soc. 26 (1982), 21–27.CrossRefGoogle Scholar
[2]Hall, M., ‘An existence theorem for latin squares’, Bull. Amer. Math. Soc. 51 (1945), 387–388.Google Scholar
[3]Hoffman, D.G. and Lindner, C.C., ‘Embeddings of Mendelsohn triple systems’, Ars Combin. 11 (1981), 265–269.Google Scholar
[4]Lindner, C.C., ‘Finite partial cyclic triple systems can be finitely embedded’, Algebra Universalis 1 (1971), 93–96.Google Scholar
[5]Lindner, C.C. and Cruse, A.B., ‘Small embeddings for partial semisymmetric and totally symmetric quasigroups’, J. London Math. Soc. 12 (1976), 479–484.Google Scholar
[6]Lindner, C.C. and Rodger, C.A., Decompositions into cycles II: cycle systems, selected surveys in combinatorial design theory (Wiley, to appear).Google Scholar
[7]Lindner, C.C. and Rodger, C.A., ‘A partial m = (2k + 1)-cycle system of order n can be embedded in an m−cycle system of order (2n + 1)m’, Discrete Math, (to appear).Google Scholar
[8]Lindner, C.C., Rodger, C.A. and Horton, J.D., ‘A small embedding for partial 4-cycle systems’, J. Combin. Math. Combin. Comput. 5 (1989), 23–26.Google Scholar
[9]Mendelsohn, N.S., ‘A natural generalisation of Steiner triple systems’, in Computers in Number Theory, Editors Atkin, A.O.L. and Birch, B.J., pp. 323–338 (Academic Press, London, 1971).Google Scholar
[10]Rodger, C.A., ‘Embedding partial Mendelsohn triple systems’, Discrete Math. 65 (1987), 187–196.CrossRefGoogle Scholar
[11]Sotteau, D., ‘Decompositions of Km, n () into cycles (circuits) of length 2k’ J. Combin. Theory Series B 29 (1981), 75–81.CrossRefGoogle Scholar
[12]Tillson, T.W., ‘A Hamilton decomposition of ,2m ≥ 8’, J. Combin. Theory Series B 29 (1980), 68–74.CrossRefGoogle Scholar
[13]Wilson, R.M., ‘Construction and uses of pairwise balanced designs’, Maths. Centre Tracts 55 (1974), 18–41.Google Scholar