On Kaleidoscope Designs Francesca Merola Roma Tre University Joint - - PowerPoint PPT Presentation

On Kaleidoscope Designs

Francesca Merola Roma Tre University

Joint work with Marco Buratti

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

x0 x3 x5 x1 x4 x2 x6

(7, 3, 1)-design (9, 3, 1)-design

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 c0,c1,. . . ,cb−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 ci there

is exactly one design of D in which the block having color ci 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

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 c0,c1,c2,c3,c4,c5,c6.

• for each pair of points x, y in V and each colour ci
• ∃! plane in D in which the line of colour ci contains x and y

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 c0,c1,c2,c3,c4,c5,c6,c7,c8,c9,c10,c11.

• for each pair of points x, y in V and each colour ci
• ∃! design in D in which the line of colour ci contains x and y

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)

h(h−1) ∈ Z and k(v−1) h(h−1) ∈ Z

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

Sch¨

• nheim (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

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}

example - a (7, 3, 1)K(19)

1 4 5 2 8 11

• in Z19 consider the 7-ple B = (0, 1, 2, 4, 5, 11, 8)
• as blocks take B =

{mB + k, m ∈ {1, 26, 212}, k ∈ Z19} = {(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 ∈ Z19}

• we have a (7, 3, 1)K(19) regular under Z19
• 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}

example - a (9, 3, 1)K(19)

• in Z19 consider the 9-ple B = (0, 1, 2, 3, 7, 16, 8, 4, 10) =

(b∞, b0, b1, b2, b3, b4, b5, b6, b7)

• as blocks take B =

{mB + k, m ∈ {1, 26, 212}, k ∈ Z19} = {(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 ∈ Z19}

• we have a (9, 3, 1)K(19) regular under Z19
• taking as lines the triples

{b∞, b0, b4}, {b∞, b1, b5}, {b∞, b2, b6}, {b∞, b3, b7} and the 8 lines of the form {bi, bi+1, bi+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}

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 (b0, b1, b2, b3, b4, b5, b6) such that

the differences in the triples {b0, b1, b3}, {b1, b2, b4}, . . . , {b6, b0, b2} satisfy specific cyclotomic conditions

• a result of Buratti and Pasotti (2009) guarantees the

existence of such a block for v > a bound β ≈ 4 800 000

• we explicitly built a design for all values of v smaller than β,

and found that no regular design exists for v = 13

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)

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
• f a (k, h, 1)K(w) for all w ∈ K implies the existence of a

(k, h, 1)K(v)

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

(7, 3, 1)K(13)

• the underlying, uncolored design is in this case a

(13, 7, 7)-design

• there are 19 072 802 such designs

(Kaski and ¨ Osterg˚ ard (2004))

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 ≈ 809 000 000
• 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 ≈ 1.38391 · 1013

• 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)!

a kaleidoscopic anniversary

the kaleidoscope was invented 200 years ago by David Brewster (1781-1868), a Scottish physicist, mathematician, astronomer, inventor, writer, historian of science and university principal