On Kaleidoscope Designs

Transcription

1 On Kaleidoscope Designs Francesca Merola Roma Tre University Joint work with Marco Buratti

2 notation (v, k, λ)-design: V = v set of points, B set of blocks, B ( V k ), B = b, such that any two points belong to exactly λ blocks x 5 x 4 x 2 x 6 x 0 x 1 (7, 3, 1)-design x3 (9, 3, 1)-design

3 definition let D be a set of (k, h, 1)-designs with b blocks the points of the designs in D belong to a given v-set V the b blocks of designs in D are colored with the same b different colors c 0,c 1,...,c b 1. We say that D is a Kaleidoscope Design of order v and type (k, h, 1), briefly a (k, h, 1)K(v) if for any two distinct points x, y of V and any color c i there is exactly one design of D in which the block having color c i contains x and y. in this talk we shall deal mostly with the case (k, h, 1) = (7, 3, 1) - Fano case and (k, h, 1) = (9, 3, 1) - affine case

4 Fano case a set D of (7, 3, 1)-designs Fano Planes with points from a v-set V the seven lines in each plane are coloured with seven different colours c 0,c 1,c 2,c 3,c 4,c 5,c 6. for each pair of points x, y in V and each colour c i! plane in D in which the line of colour c i contains x and y

5 affine case a set D of (9, 3, 1)-designs with points from a v-set V the 12 lines in each plane are coloured with 12 different colours c 0,c 1,c 2,c 3,c 4,c 5,c 6,c 7,c 8,c 9,c 10,c 11. for each pair of points x, y in V and each colour c i! design in D in which the line of colour c i contains x and y

6 necessary conditions the underlying - uncoloured - structure is a (v, k, b) design each pair of points x and y V appear together in b small designs so the usual admissibility conditions for designs apply for the (7, 3, 1) case, we have v 1 (mod 6) for the (9, 3, 1) case, we have v 1, 3 (mod 6) for (k, h, 1)K(v) we have v(v 1) k(v 1) h(h 1) Z and h(h 1) Z

7 coloured designs a Kaleidoscope Design is a special instance of the very broad notion of coloured design/edge-coloured graph decomposition various definitions with more/less generality important asymptotic result by Lamken and Wilson (2000) studied by Colbourn and Stinson (1988); Caro, Roditty and Schönheim (1995,1997,2002); Adams, Bryant and Jordon (2006) many concepts (e.g. perfect cycle systems, whist tournaments, nested triple systems,... ) can be rephrased in this setting

8 repetitions if a (v, k, 1) design exists, by replicating each block b times and shifting cyclically the colours one has a Kaleidoscope design for instance in the Fano case for v=7, we just repeat the one block (0, 1, 2, 3, 4, 5, 6) seven times {0, 1, 3},{1, 2, 4},{2, 3, 5},{3, 4, 6},{4, 5, 0},{5, 6, 1},{6,0,2} {0, 1, 3},{1, 2, 4},{2, 3, 5},{3, 4, 6},{4, 5, 0},{5, 6, 1},{6,0,2} {0, 1, 3},{1, 2, 4},{2, 3, 5},{3, 4, 6},{4, 5, 0},{5, 6, 1},{6,0,2} {0, 1, 3},{1, 2, 4},{2, 3, 5},{3, 4, 6},{4, 5, 0},{5, 6, 1},{6,0,2} {0, 1, 3},{1, 2, 4},{2, 3, 5},{3, 4, 6},{4, 5, 0},{5, 6, 1},{6,0,2} {0, 1, 3},{1, 2, 4},{2, 3, 5},{3, 4, 6},{4, 5, 0},{5, 6, 1},{6,0,2} {0, 1, 3},{1, 2, 4},{2, 3, 5},{3, 4, 6},{4, 5, 0},{5, 6, 1},{6,0,2}

9 example - a (7, 3, 1)K(19) in Z 19 consider the 7-ple B = (0, 1, 2, 4, 5, 11, 8) as blocks take B = {mb + k, m {1, 2 6, 2 12 }, k Z 19 } = {(0, 1, 2, 4, 5, 11, 8) + k, (0, 7, 14, 9, 16, 1, 18) + k, (0, 11, 3, 6, 17, 7, 12) + k; k Z 19 } we have a (7, 3, 1)K(19) regular under Z 19 taking as lines the triples in position i, i + 1, i + 3 (ind mod 7) and colouring the i-th triple with the i-th colour so for B we have {0, 1, 4},{1, 2, 5},{2, 4, 11},{4, 5, 8},{5, 11, 0},{11, 8, 1},{8, 0, 2}

10 example - a (9, 3, 1)K(19) in Z 19 consider the 9-ple B = (0, 1, 2, 3, 7, 16, 8, 4, 10) = (b, b 0, b 1, b 2, b 3, b 4, b 5, b 6, b 7 ) as blocks take B = {mb + k, m {1, 2 6, 2 12 }, k Z 19 } = {(0, 1, 2, 3, 7, 16, 8, 4, 10) + k, (0, 7, 14, 2, 11, 17, 18, 9, 13) + k, (0, 11, 3, 14, 1, 5, 12, 6, 15) + k; k Z 19 } we have a (9, 3, 1)K(19) regular under Z 19 taking as lines the triples {b, b 0, b 4 }, {b, b 1, b 5 }, {b, b 2, b 6 }, {b, b 3, b 7 } and the 8 lines of the form {b i, b i+1, b i+3 } (ind mod 8) and colouring each of these 12 lines with a different color {0, 1, 16}, {0, 2, 8}, {0, 3, 4}, {0, 7, 10}, {1, 2, 7}, {2, 3, 16}, {3, 7, 8},{7, 16, 4},{16, 0, 10},{8, 4, 1},{4, 10, 2},{10, 1, 3}

11 existence results for the Fano case we used difference methods, and considered the case in which v is prime or a prime power Theorem A regular (7, 3, 1)K(v) exists for all v 1 (mod 6), whenever v is prime or a prime power, v 13. look for a starting block (b 0, b 1, b 2, b 3, b 4, b 5, b 6 ) such that the differences in the triples {b 0, b 1, b 3 }, {b 1, b 2, b 4 },..., {b 6, b 0, b 2 } satisfy specific cyclotomic conditions a result of Buratti and Pasotti (2009) guarantees the existence of such a block for v > a bound β we explicitly built a design for all values of v smaller than β, and found that no regular design exists for v = 13

12 recursive existence results this result can be applied recursively with the help of difference matrices to give Theorem A regular (7, 3, 1)K(v) exists for all v whose prime power factors are all 1 (mod 6) (but not = 13)

13 PBDs a (v, K, λ)-pairwise Balanced Design, or (v, K, λ)-pbd is a pair (V, B), where V is a v-set of points and B is a set of blocks, subsets of V with B K B B s.t. each pair of points of V is contained in λ blocks B B when K = {k}, this gives a (v, k, λ)-design. PBDs are very useful in constructing ordinary designs and also in building Kaleidoscope designs: the existence of a (v, K, 1)-PBD together with the existence of a (k, h, 1)K(w) for all w K implies the existence of a (k, h, 1)K(v)

14 PBDs existence results we can make use of the following result on PBDs Theorem (Mullin and Stinson (1987), Greig (1999)) Let K be the set of prime powers 1 (mod 6); then, for all v 1 (mod 6) with at most 22 exceptions, a (v, K, 1) PBD exists. a (7, 3, 1)K(v) can exist only if v 1 (mod 6), and we have proved existence for all for all orders that are prime powers 1 (mod 6) (except for 13!!!) if we build a (7, 3, 1)K(13), this result together with our existence result implies the existence of a (7, 3, 1)K(v) for all admissible values with at most 22 exceptions even without a (7, 3, 1)K(13), the result of Mullin and Stinson allows us to prove the existence of a (7, 3, 1)K(v) for many non prime power values of v

15 (7, 3, 1)K(13) the underlying, uncolored design is in this case a (13, 7, 7)-design there are such designs (Kaski and Östergård (2004))

16 existence results for the affine and general case we could use similar methods to obtain existence results in the affine case: via difference methods and the use of cyclotomic conditions, once more using Buratti and Pasotti (2009) we can state that Theorem A regular (9, 3, 1)K(v) exists for all v 1 (mod 6), whenever v is prime or a prime power, v > than a bound β. the bound in this case is β harder (but probably possible) to consider all the values smaller than β - once more, regular for v = 13 were we trying to apply the same methods to (13, 3, 1)K(v), the bound would be β also we cannot rely on the results on PBDs; as in the Fano case, a non regular (9, 3, 1)K(13) might exist, but not a (9, 3, 1)K(7)!

17 a kaleidoscopic anniversary the kaleidoscope was invented 200 years ago by David Brewster ( ), a Scottish physicist, mathematician, astronomer, inventor, writer, historian of science and university principal