Constructing R-sequencings and terraces for groups of even order

M. A. Ollis

Algebra and discrete mathematics, Tome 20 (2015) no. 2, pp. 297-316.

Résumé

The problem of finding R-sequencings for abelian groups of even orders has been reduced to that of finding R$^*$-sequencings for abelian groups of odd orders except in the case when the Sylow 2-subgroup is a non-cyclic non-elementary-abelian group of order 8. We partially address this exception, including all instances when the group has order $8t$ for $t$ congruent to 1, 2, 3 or 4 $(\operatorname{mod} 7)$. As much is known about which odd-order abelian groups are R$^*$-sequenceable, we have constructions of R-sequencings for many new families of abelian groups. The construction is generalisable in several directions, leading to a wide array of new R-sequenceable and terraceable non-abelian groups of even order.

Détail

Keywords: 2-sequencing, Bailey's Conjecture, R-sequencing, terrace.

Bibliographie

[1] A. Ahmed, M. I. Azimli, I. Anderson and D. A. Preece, “Rotational terraces from rectangular arrays”, Bulletin Inst. Combinatorics Appl., 63 (2011), 4–12 | MR | Zbl

[2] B. A. Anderson and E. C. Ihrig, “All groups of odd order have starter-translate 2-sequencings”, Australas. J. Combin., 6 (1992), 135–146 | MR | Zbl

[3] B. A. Anderson and E. C. Ihrig, “Symmetric sequencings of non-solvable groups”, Congr. Numer., 93 (1993), 73–82 | MR | Zbl

[4] R. A. Bailey, “Quasi-complete Latin squares: construction and randomization”, J. Royal Statist. Soc. Ser. B, 46 (1984), 323–334 | MR | Zbl

[5] R. J. Friedlander, B. Gordon and M. D. Miller, “On a group sequencing problem of Ringel”, Congr. Numer., 21 (1978), 307–321 | MR

[6] GAP group, GAP — Groups, Algorithms, and Programming, Version 4, 1999

[7] B. Gordon, “Sequences in groups with distinct partial products”, Pacific J. Math., 11 (1961), 1309–1313 | DOI | MR | Zbl

[8] P. Headley, “R-sequenceability and R$^*$-sequenceability of abelian 2-groups”, Discrete Math., 131 (1994), 345–350 | DOI | MR | Zbl

[9] A. D. Keedwell, “On the $R$-sequenceability and $R_{h}$-sequenceability of groups”, Ann. Discrete Math., 18 (1983), 535–548 | MR | Zbl

[10] G. N. Martin and M. A. Ollis, “R-sequencings and strong half-cycles from narcissistic terraces”, Australas. J. Combin, 63 (2015), 346–367 | MR | Zbl

[11] M. A. Ollis, “Sequenceable groups and related topics”, Electron. J. Combin., DS10 (2002), 34 pp., (updated 2013)

[12] M. A. Ollis, “On terraces for abelian groups”, Discrete Math., 305 (2005), 250–263 | DOI | MR | Zbl

[13] M. A. Ollis, “A note on terraces for abelian groups”, Australas. J. Combin., 52 (2012), 229–234 | MR | Zbl

[14] M. A. Ollis and D. T. Willmott, “On twizzler, zigzag and graceful terraces”, Australas. J. Combin., 51 (2011), 243–257 | MR | Zbl

[15] M. A. Ollis and D. T. Willmott, “An extension theorem for terraces”, Electron. J. Combin., 20 (2013), 10 pp. | MR | Zbl

[16] M. A. Ollis and D. T. Willmott, “Constructions for terraces and $R$-sequencings, including a proof that Bailey's Conjecture holds for abelian groups”, J. Combin. Des., 23 (2015), 1–17 | DOI | MR | Zbl

[17] L. J. Paige, “Complete mappings of finite groups”, Pacific J. Math., 1 (1951), 111–116 | DOI | MR | Zbl

[18] C.-D. Wang and P. A. Leonard, “On R$^*$-sequenceability and symmetric harmoniousness of groups”, J. Combin. Des., 2 (1994), 71–78 | DOI | MR | Zbl

[19] C.-D. Wang and P. A. Leonard, “On R$^*$-sequenceability and symmetric harmoniousness of groups II”, J. Combin. Des., 3 (1995), 313–320 | DOI | MR | Zbl

[20] C.-D. Wang and P. A. Leonard, “More on sequences in groups”, Australas. J. Combin., 21 (2000), 187–196 | MR | Zbl

[21] E. J. Williams, “Experimental designs balanced for the estimation of residual effects of treatments”, Aust. J. Scient. Res. A, 2 (1949), 149–168 | MR