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Difference Families in Z2 d +1⊕Z2 d +1 and Infinite Translation Designs in Z⊕Z

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Abstract

We analyse 3-subset difference families of Z2 d +1⊕Z2 d +1 arising as reductions (mod 2d+1) of particular families of 3-subsets of Z⊕Z. The latter structures, namely perfect d-families, can be viewed as 2-dimensional analogues of difference triangle sets having the least scope. Indeed, every perfect d-family is a set of base blocks which, under the natural action of the translation group Z⊕Z, cover all edges {(x,y),(x′,y′)} such that |x−x′|, |y−y′|≤d. In particular, such a family realises a translation invariant (G,K3)-design, where V(G)=Z⊕Z and the edges satisfy the above constraint. For that reason, we regard perfect families as part of the hereby defined translation designs, which comprise and slightly generalise many structures already existing in the literature. The geometric context allows some suggestive additional definitions. The main result of the paper is the construction of two infinite classes of d-families. Furthermore, we provide two sporadic examples and show that a d-family may exist only if d≡0,3,8,11 (mod 12).

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References

  1. Abel, R.J.L.: Difference families. In: CRC Handbook of Combinatorial Designs (C.J. Colbourn and J.H. Dinitz eds.), CRC Press, Boca Raton, FL, 1996, pp. 270–287

  2. Abel, J., Buratti, M.: Some progress on (v,4,1) difference families, J. Combin. Theory A 106, 2004, pp. 59–75

    Google Scholar 

  3. Alspach, B., Gavlas, H.: Cycle decompositions of K n and K n –I, J. Combin. Theory B 81, 2001, pp. 77–99

  4. Anderson, I.: Combinatorial Designs and Tournaments, Oxford Lecture Series in Mathematics and its Applications, vol. 6, Clarendon Press, Oxford, 1997

  5. Beth, T., Jungnickel, D., Lenz, H.: Design theory, Cambridge University Press, Cambridge, 1999

  6. Buratti, M.: A description of any regular or 1-rotational design by difference methods, Booklet of the abstracts of Combinatorics 2000, http://www.mat.uniroma1.it/combinat/gaeta/index.html.

  7. Buratti, M., Del Fra, A.: Cyclic Hamiltonian cycle systems of the complete graph. In honour of Zhu Lie, Discr. Math. 279, 2004, pp. 107–119

  8. Buratti, M., Del Fra, A.: Existence of cyclic k-cycle systems of the complete graph, Discr. Math. 261, 2003, pp. 113–125

  9. Chung, F.R.K., Salehi, J.A., Wei, V.K.: Optical orthogonal codes: design, analysis and application, IEEE Trans. Inform. Theory 35, 1989, pp. 595–604

    Google Scholar 

  10. Colbourn, C.J.: Difference triangle sets. In: CRC Handbook of Combinatorial Designs (C.J. Colbourn and J.H. Dinitz eds.), CRC Press, Boca Raton, FL, 1996, pp. 313–317

  11. Fu, H., Wu, S.: Cyclically decomposing the complete graph into cycles, Discr. Math. 282, 2004, pp. 267–273

    Google Scholar 

  12. Heinrich, K.: Graph decompositions and designs. In: CRC Handbook of Combinatorial Designs (C.J. Colbourn and J.H. Dinitz eds.), CRC Press, Boca Raton, FL, 1996, pp. 361–366

  13. Hoffman, D.G., Lindner, C.C., Rodger, C.A.: On the construction of odd cycle systems, J. Graph Theory 13, 1989, pp. 417–426

    Google Scholar 

  14. Kotzig, A. Decompositions of a complete graph into 4k-gons, (Russian) Mat.-Fyz. Časopis Sloven. Akad. Vied 15, 1965, pp. 229–233

  15. Lindner, C.C. Rodger, C.A.: Decomposition into cycles II: cycle systems. In: Contemporary Design Theory: a collection of surveys (J.H. Dinitz, D.R. Stinson eds.), John Wiley and Sons, New York, 1992, pp. 325–369

  16. Rodger, C.A.: Cycle Systems. In: CRC Handbook of Combinatorial Designs (C.J. Colbourn and J.H. Dinitz eds.), CRC Press, Boca Raton, FL, 1996, pp. 266–270

  17. Rosa, A.: On cyclic decompositions of the complete graph into (4m+2)-gons, Mat.-Fyz. Časopis Sloven. Akad. Vied 16, 1966, pp. 349–352

    Google Scholar 

  18. Rosa, A.: On decompositions of a complete graph into 4k-gons, (Russian) Mat. Časopis Sloven. Akad. Vied 17, 1967, pp. 242–246

    Google Scholar 

  19. Rosa, A.: On the cyclic decomposition of the complete graph into polygons with odd number of edges, (Slovak) Časopis Pěst. Math. 91, 1966, pp. 53–63

    Google Scholar 

  20. Šajna, M.: Cycle decompositions III. Complete graphs and fixed length cycles, J. Combin. Designs 10, 2002, pp. 27–78

    Google Scholar 

  21. Vietri, A.: Cyclic k-cycle systems of order 2kn+k; a solution of the last open cases, J. Combin. Designs 12, 2004, pp. 299–310

    Google Scholar 

  22. Vietri, A.: Nice lists of differences, and their connections with algebraic properties of unitary polynomials, Australas. J. Combin. 37, 2007, pp. 171–185

    Google Scholar 

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  1. Dipartimento Me.Mo.Mat, Università Roma1, via A. Scarpa 16, 00161, Roma, Italia

    Andrea Vietri

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  1. Andrea Vietri
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Vietri, A. Difference Families in Z2 d +1⊕Z2 d +1 and Infinite Translation Designs in Z⊕Z. Graphs and Combinatorics 23, 111–121 (2007). https://doi.org/10.1007/s00373-006-0685-9

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