# New Results in Part VI

This page contains new results in the area of combinatorial designs that have occurred since the publication of the Handbook of Combinatorial Designs, Second Edition in November 2006. The results here would be contained in Part VI of the Handbook.

Last edited 3/10/13

Page 346, Conjecture 6.8: The Hall-Paige conjecture has been proved. J. N. Bray, A. B. Evans (anthony.evans@wright.edu), and S. Wilcox (swilcox@fas.harvard.edu ). 2/08

Page 351, Theorem 6.48.   The converse of this theorem has been proven for abelian groups.  It is called the Friedlander, Gordon, Miller Conjecture.  Specifically, the theorem that is proven is the following:  If G is a finite abelian group, then G is R-sequenceable if the Sylow 2-subgroup either is trivial, or the Sylow 2-subgroup is non-trivial and non-cyclic.  Reference:  B. Alspach, D.Kreher and A. Pastine, The Friedlander-Gordon-Miller conjecture is true, Australasian J. of Combinatorics 67 (2017), 11-24.

Page 360, Table 9.36: The enumeration of Costas Arrays of orders 27 is complete. C(27)=204 and c(27)=29. Also, there is now an excellent website devoted to information concerning Costas Arrays, the URL is http://www.costasarrays.org/ . Scott Rickard (scott.rickard@ucd.ie). 5/08

Page 360, Table 9.36: The enumeration of Costas Arrays of orders 28 and 29 is complete. C(28)=712 and c(28)=89; C(29)=164 and c(29)=23. 5/12

Page 368, Theorem 11.29. (44,5, λ) coverings are now known for λ = 17 and λ ≡ 13 (mod 20). Julian Abel (r.j.abel@unsw.edu.au). 8/09

Page 375, Theorem 12.23. Solutions to the Oberwolfach problem for orders from 18 through 40 have been found. See Deza, Franek, Hua, Mezka and Rosa, "Solutions to the Oberwolfach problem for orders 18 to 40, JCMCC 74 (2010), pp 95-102. The solutions can be found online at http://optlab.mcmaster.ca/~oberwolfach/ . 8/10.

Page 375, Theorem 12.23. There is a solution to all Oberwolfach problems (including when λ > 1) with two table lengths. Ref: A complete solution to the two-table Oberwolfach problems, Tommaso Traetta, (preprint). Tommaso Traetta (tommaso.tr@gmail.com). 5/12

Page 382. Theorem 12.80. The existence problem for 1-rotational k-cycle systems of K_n has been completely solved. The result is as follows: There exists a 1-rotational k-cycle system of order n if and only if k is odd and n is admissible provided that n is not 1 (mod 2k) when k is a prime and n is neither 15 nor 21 (mod 24) when k=3. The references are [387] and the paper S.L. Wu and M. Buratti, A complete solution to the existence problem for 1-rotational k-cycle systems of K_v, J. Combin. Des. 17 (2009), 283-293. Marco Buratti (buratti@dmi.unipg.it), June 2012.

Page 397, Theorem 16.29: The bound that q > {k \choose 2}^{k(k-1)} can be improved to q > ({k \choose 2}^{2k})/ \gcd({k \choose 2},\lambda)^{2k-2}). Anita Pasotti (anita.pasotti@ing.unibs.it). 4/08

Page 406. Theorem 16.77. Much progress on the case v=1 (mod 6) has been made in the paper S. Bonvicini, M. Buratti, G. Rinaldi and T. Traetta, Some progress on 1-rotational Steiner triple systems, Des. Codes Cryptogr. 62 (2012), 63-78. The main result can be summarized as follows: For the existence of a 1-rotational STS(v) with v ≡ 1 (mod 6) it is necessary that v ≡ 1 or 19 (mod 24). The case v ≡ 19 (mod 24) has been completely solved: in this case a 1-rotational STS(v) cannot exist only when v=6p1p2...pt+1 with the p_i's pairwise distinct primes congruent to 5 (mod 6). The case v ≡ 1 (mod 24) is quite complicated. However the existence is uncertain only when v has the following very special form: v=(p3-p)n+1 ≡ 1 (mod 96) with p a prime, n \not\equiv 0 (mod 4) and the odd part of v-1 which is square-free and without prime factors ≡ 1 (mod 6). Marco Buratti (buratti@dmi.unipg.it), June 2012.

Page 417, Theorem 17.52: V(9,t) vectors exist for all for t > 8, q = 9t+1 a prime power, except possibly for q = 56. (K. Chen, Z. Cao and R. Wei, Existence of V(9,t) vectors, JCMCC 55 (2005), 209-221).

Page 436. Monotonic directed designs were introduced by Chu, Colbourn and Golomb to construct difference triangle sets. Ge, Huang and Miao developed various constructions for monotonic directed designs. Click here to see some small designs which are essential to the establishment of the necessary and sufficient conditions for the existence of a monotonic directed design with block size 3, and with block size 4 having two definite exceptions and six possible exceptions.

Page 478, near Theorem 24.10: If G is a simple graph with e edges and degeneracy d (that is the largest minimal degree among the minimal degrees of all the subgraphs of G), then there exists an elementary abelian (Kq,G)-design for every prime power q such that e{2d+2} < q ≡ 1 (mod 2e). Anita Pasotti (anita.pasotti@ing.unibs.it). 4/08

Page 481, Theorem 24.45(4): There exists a (Kn, G22)-designs for n=27, 135, 162, and 216. Emre Kolotoglu (ekolotog@fau.edu), 2/12.

Page 502, Table 29.28: There was an error in the original work and it turns out that ν(6,10) = 3. Indeed there exists an H3*(6,10). The Howell design can be found here. The reference is J. Dinitz and G. Warrington, The spectra of certain classes of Room frames: The last cases, Electron. J. Combin. 17 (2010), #R74, 13 pages.

Page 533, Table 35.46: There are 6,855,400,728 inequivalent MTS(13,1). Reference: The Mendelsohn Triple Systems of Order 13, by Khatirinejad, Ostergard and Popa, preprint. 7/12.

Page 569, Theorem 44.4: In addition, M(n,d) is not equal to n!/(d-1)!-1. (J. Quistorff, Electron. J. Combin., 13 (2006), #A1, www.combinatorics.org/Volume_13/PDF/v13i1a1.pdf). Peter Dukes (dukes@uvic.ca), 11/07

Page 571, Table 44.32: M(12,11) ≥ 112. This is a result of Ingo Janiszczak and Reiner Staszewski. Click here to see the construction. Ingo Janiszczak (ingo@iem.uni-due.de), 1/13.

Page 575, Remark 46.6 and Theorem 46.7: Colbourn [529] developed a strategy for group testing when the clones are linearly ordered and the positive clones form a consecutive subset of the set of all clones. Jimbo and his collaborators improved Colbourn's strategy by considering the error detecting and correcting capability of group testing. In order to correct false negative or false positive clones in the pool outcomes, Momihara and Jimbo suggest the investigation of block sequences of maximum packings MP(t,k,v) which contains the blocks exactly once such that the collection of all blocks together with all unions of two consecutive blocks of this sequence forms an error correcting code with minimum distance d. Such a sequence is usually called a block sequence with consecutive unions having minimum distance d, and denoted by BSCU(t,k,v|d). Ge, Miao and Zhang ("On block sequences of Steiner quadruple systems with error correcting consecutive unions", SIAM J. Discrete Math., to appear) showed that the necessary conditions for the existence of BSCU(3,4,v|4)'s of Steiner quadruple systems, namely, v ≡ 2,4 mod(6) and v ≥ 4, are also sufficient except for v = 8,10.
Some small useful cyclic sequence of blocks with consecutive unions can be found here. This is essentially the appendix to the paper by Ge, Miao and Zhang mentioned above. Ying Miao (miao@sk.tsukuba.ac.jp), 3/09

Page 587, Theorem 50.29 There exists a Room square of side 67 containing a subsquare of side 21. See: J. Dinitz and G. Warrington, The spectra of certain classes of Room frames: The last cases, Electron. J. Combin. 17 (2010), #R74, 13 pages.

Page 588, Theorem 50.34(2) There exists a Room frame of type 144. See: J. Dinitz and G. Warrington, The spectra of certain classes of Room frames: The last cases, Electron. J. Combin. 17 (2010), #R74, 13 pages.

Page 588, Theorem 50.35(1) There exists a Room frame of type 219181. See: J. Dinitz and G. Warrington, The spectra of certain classes of Room frames: The last cases, Electron. J. Combin. 17 (2010), #R74, 13 pages.

Page 624, Table 55.30. DS(35) = 2138089212789. (V. Linja-aho and P. R. J. Ostergard, "Classification of starters" to appear in Journal of Combinatorial Mathematics and Combinatorial Computing (JCMCC). Vesa Linja-aho (vesa.linja-aho@tkk.fi), 5/09

Page 625, Table 55.36. There are corrections to several values in this table. They are:
There are 2 distinct strong starters for n=7 (not 1).
There are 8 distinct strong starters for n=13 (not 4).
There are 6660 distinct strong starters for n=21 (not 6600).
There are 1201626 distinct strong starters for n=27 (not 1249650).
There are 66757 inequivalent strong starters for n=27 (not 69425).
In addition, the number of strong starters in cyclic groups has been computed up to n = 37. All of this can be found V. Linja-aho and P. R. J. Ostergard, "Classification of starters" to appear in Journal of Combinatorial Mathematics and Combinatorial Computing (JCMCC). Vesa Linja-aho (vesa.linja-aho@tkk.fi), 5/09

Page 627. Definition 55.42. The definition can be extended to non-abelian groups. Also, in the paper by Bonvicini et al.(S. Bonvicini, M. Buratti, G. Rinaldi and T. Traetta, Some progress on 1-rotational Steiner triple systems, Des. Codes Cryptogr. 62 (2012), 63-78.) it is observed that as a consequence of a result by B.A. Anderson and E.C. Ihrig, the following holds: Every solvable group G with exactly one involution admits an even starter. Marco Buratti (buratti@dmi.unipg.it), June 2012.

Page 634, Theorem 57.11: Super-simple (v,5,2) BIBDs exist for v = 115 and 135 (R.J.R. Abel and F.E. Bennett, Discrete Appl. Math. 156 (2008), 780-793. Julian Abel (rjabel@unsw.edu.au), 3/08

Page 656, Table 62.26. An exhaustive search has shown that there is no 9 x 9 Tuscan-2 square. Daniele Bartoli (daniele275@gmail.com), 1/12

Page 660, Theorem 63.33 and Remark 63.34: In 1987, Hartman showed that the necessary condition v ≡ 4 or 8 (mod 12) for the existence of a resolvable SQS(v) is also sufficient for all values of v, with 23 possible exceptions. These last 23 undecided orders were removed by Ji and Zhu in 2005, where the concept of resolvable H-designs was introduced to construct resolvable SQSs.

Zhang and Ge ("Existence of resolvable H-designs with group sizes 2,3,4 and 6", Designs, Codes and Cryptography, to appear) showed that: The necessary conditions gn ≡ 0 (mod 4), g(n-1)(n-2) ≡ 0 (mod 3) and n ≥ 4 for the existence of a resolvable H-design of type g^{n} are also sufficient for each g = 1,2,3,5,6,7,9,10,11 (mod 12), and also sufficient for each g ≡ 4,8 (mod 12) with two possible exceptions n=73,149. Direct constructions for RH(419) and RH(441) can be found here. This is essentially the appendix to the paper by Zhang and Ge mentioned above. As a consequence, they also show that the necessary conditions for the existence of a resolvable G-design of type g^n are also sufficient. Xiande Zhang (xdzhangzju@163.com). 9/09