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Group Partition, Factorization and the Vector Covering Problem

Published online by Cambridge University Press:  20 November 2018

M. Herzog  and
J. Schönheim
M. Herzog
Affiliation:
Tel-Aviv University, Tel-Aviv, Israel
J. Schönheim
Affiliation:
Tel-Aviv University, Tel-Aviv, Israel
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The covering problem. Let Si(i = 1, 2,…,n) be given sets containing mi elements respectively and let

1

be their cartesian product. The elements of S(n) will be called vectors. The vector (x1 x2,…, xn) covers (y1 y2,…, yn) if xi =yi for at least n—1 values of i. A subset M of S(n) is said to be a covering (perfect covering) of S(n) if each member of S(n) is covered by at least (exactly) one member of M. A covering M is said to be linear if the sets Si are groups Gi and M is a subgroup of G(n) = S(n) Denote by σ(n; m1 m2,…, mn) the value of min |M| when M runs through all coverings of S(n) and by σ(n; m1 m2,…, mn) the value of min |M| when the sets Si are given groups Gi and M runs through all linear coverings of G(n).

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 15 , Issue 2 , June 1972 , pp. 207 - 214
Copyright
Copyright © Canadian Mathematical Society 1972

References

1. Taussky, O. and Todd, Y., Covering theorems for groups, Ann. Polon. Math. 21 (1949), 303-308.Google Scholar
2. Kalbfleisch, J. G. and Stanton, R. G., A combinatorial problem in matching, J. Londo. Math. Soc. 44 (1969), 60-64.Google Scholar
3. Zaremba, S. K., Covering problems concerning Abelian Groups, J. Londo. Math. Soc. 27 (1952), 242-246.Google Scholar
4. Losey, G., Note on a theorem of Zaremba, J. Comb. Theory, 6 (1969), 208-209.Google Scholar
5. Zaremba, S. K., A covering theorem for Abelian groups, J. Londo. Math. Soc. 26 (1950), 71-72.Google Scholar
6. Sands, A. D., Factorizations of cyclic groups, Proc. Coll. on Abelian Groups, Akademiai Kiado Budapest (1964), 139-146.Google Scholar
7. Baer, R., PartiMonen endlicher Gruppen, Math. Z. 75 (1961), 337.Google Scholar
8. Lindstrom, B., On group and nongroup perfect codes in q symbols, Math. Scand. 25 (1969), 145-158.Google Scholar