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Embedding partial Latin squares into Latin squares with many orthogonal mates Emine ule Yazc Ko University Joint work with D. Donovan and M. Grannell TUBITAK 116F166 LATIN SQUARES Latin square of order n is an n x n array on the

  1. Embedding partial Latin squares into Latin squares with many orthogonal mates Emine Şule Yazıcı Koç University Joint work with D. Donovan and M. Grannell TUBITAK 116F166

  2. LATIN SQUARES ◼ Latin square of order n is an n x n array on the set of symbols {1,2,...,n}, such that each row and column of the array contains each symbol exactly once. . 1 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 1 3 4 5 6 7 8 9 1 2 4 5 6 7 8 9 1 2 3 Latin Square 5 6 7 8 9 1 2 3 4 of order 9 6 7 8 9 1 2 3 4 5 7 8 9 1 2 3 4 5 6 8 9 1 2 3 4 5 6 7 9 1 2 3 4 5 6 7 8

  3. Mutually orthogonal latin squares ◼ The latin squares L 1 , L 2 ,...,L t are said to be mutually orthogonal if for 1 ≤ a≠b ≤t, L a and L b are orthogonal. ◼ Latin squares L a and L b of order n are said to be orthogonal if for each (x,y)  {1,2,...n}x{1,2,...,n}, there exists one order pair (i,j) such that the cell (i,j) of L a contains the symbol x and the cell (i,j) of L b contains the symbol y

  4. A pair of mutually orthogonal latin squares of order 5 1 2 3 4 5 1 1 2 2 3 3 4 4 5 5 2 3 4 5 1 3 3 4 4 5 5 1 1 2 2 3 4 5 1 2 5 5 1 1 2 2 3 3 4 4 4 5 1 2 3 2 2 3 3 4 4 5 5 1 1 5 1 2 3 4 4 4 5 5 1 1 2 2 3 3 All ordered pairs (x,y)  {1,2,...,5}x{1,2,...,5} appears once in the superimposed latin square

  5. EMBEDDINGS OF LATIN SQUARES

  6. Embeddings of Latin Squares ◼ A latin square L of order n is embedded in a latin square K of order m if K contains L as a subsquare

  7. Example 1 2 3 4 5 6 7 8 9 3 1 2 5 6 4 8 9 7 2 3 1 6 4 5 9 7 8 7 8 9 1 2 3 4 5 6 8 9 7 2 3 1 5 6 4 9 7 8 3 1 2 6 4 5 4 5 6 7 8 9 1 2 3 5 6 4 8 9 7 2 3 1 6 4 5 9 7 8 3 1 2 A latin square of order 3 embedded in a latin square of order 9

  8. A pair of orthogonal latin squares of order 3 embedded in a pair of orthogonal latin squares of order 9 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 2 3 1 5 6 4 8 9 7 3 1 2 6 4 5 9 7 8 3 1 2 6 4 5 9 7 8 2 3 1 5 6 4 8 9 7 7 8 9 1 2 3 4 5 6 4 5 6 7 8 9 1 2 3 8 9 7 2 3 1 5 6 4 6 4 5 9 7 8 3 1 2 9 7 8 3 1 2 6 4 5 5 6 4 8 9 7 2 3 1 4 5 6 7 8 9 1 2 3 7 8 9 1 2 3 4 5 6 5 6 4 8 9 7 2 3 1 9 7 8 3 1 2 6 4 5 6 4 5 9 7 8 3 1 2 8 9 7 2 3 1 5 6 4

  9. Embeddings of Mutually Orthogonal Latin Squares ◼ (1986) A pair of orthogonal latin squares of order n can be embedded in a pair of orthogonal latin squares of all orders t≥3n.

  10. Partial Latin Squares ◼ A partial Latin square is an n × n array with entries chosen from a set of n symbols such that each symbol occurs at most once in each row and at most once in each column. ◼ A partial Latin square can be thought of as a subset of a Latin square . 1 4 Partial Latin square of order 4 3 4 3 2 2 3

  11. Embedding partial Latin Squares 0 1 2 3 4 5 6 3 4 5 6 0 1 2 6 0 1 2 3 4 5 2 3 4 5 6 0 1 5 6 0 1 2 3 4 1 2 3 4 5 6 0 4 5 6 0 1 2 3

  12. Embeddings of partial Latin Squares ◼ Evan’ s Theorem (1960) : A partial Latin square of order n can always be embedded in some Latin square of order t≥2n.

  13. Embeddings of Mutually Orthogonal Partial Latin Squares ◼ When can k mutually orthogonal partial latin squares embedded in (completed to) a set of mutually orthogonal Latin squares?

  14. Examples 1 2 1 2 1 2 3 4 4 3 5 1 4 1 3 3 mutually orthogonal partial latin squares

  15. Examples 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 2 3 4 5 1 3 4 5 1 2 4 5 1 2 3 3 4 5 1 2 5 1 2 3 4 2 3 4 5 1 4 5 1 2 3 2 3 4 5 1 5 1 2 3 4 5 1 2 3 4 4 5 1 2 3 3 4 5 1 2 3 partial mutually orthogonal latin squares embedded in 3 mutually orthogonal latin squares of order 5

  16. Situation so far ◼ Lindner (1976) : A set of k mutually orthogonal partial Latin squares can always be finitely embedded in k mutually orthogonal Latin squares. ◼ Hilton, Rodger, Wojciechowski (1992): Formulated some necessary conditions for a pair of partial orthogonal Latin squares to be extended to a pair of Latin squares.

  17. Situation so far ◼ Jenkins (2005): A partial Latin square of order n can be embedded in a Latin square of order 4n 2 which has an orthogonal mate.

  18. ▪ Donovan, Yazici (2014) A pair of orthogonal partial Latin squares can always be embedded in a pair of orthogonal Latin squares of polynomial order with respect to the order of the partial squares

  19. Embeddings of 2 orthogonal partial Latin squares (2014) ◼ A pair of partial orthogonal latin squares of order n can be embedded in a pair of orthogonal latin squares of order m where m is at most 16n 4 ◼ A pair of orthogonal partial latin squares of order n can be embedded in a pair of orthogonal latin squares of all orders m≥48n 4 .

  20. Embedding with many mates 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 3 4 5 6 0 1 2 1 2 3 4 5 6 0 2 3 4 5 6 0 1 6 0 1 2 3 4 5 2 3 4 5 6 0 1 4 5 6 0 1 2 3 2 3 4 5 6 0 1 3 4 5 6 0 1 2 6 0 1 2 3 4 5 5 6 0 1 2 3 4 4 5 6 0 1 2 3 1 2 3 4 5 6 0 1 2 3 4 5 6 0 5 6 0 1 2 3 4 3 4 5 6 0 1 2 4 5 6 0 1 2 3 6 7 1 2 3 4 5 5 6 0 1 2 3 4

  21. Embedding with many mates ◼ First embed the partial Latin square into a Latin square of order n 0 0 1 1 3 4 2 2 0 0 1 3 4 3 4 0 2 1 4 3 2 1 0 1 2 4 0 3 B

  22. Embedding with many mates ◼ Then we take a set of t mutually orthogonal Latin squares of order n F 1 F 2 F 3 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 1 2 3 4 0 2 3 4 0 1 3 4 0 1 2 2 3 4 0 1 4 0 1 2 3 1 2 3 4 0 3 4 0 1 2 1 2 3 4 0 4 0 1 2 3 4 0 1 2 3 3 4 0 1 2 2 3 4 0 1

  23. Embedding with many mates ◼ X k ={((p,r),(q,c),[F k (F 1 ( p ,r),q),F k (F 1 (p, q ),c)]} ◼ B*= {((p,r),(q,c),[F 1 (p,q),B(F 1 ( p ,r),c)]} ◼ Let pq=F 1 (p,q)

  24. B* (q,c) (0,c) (0,r) (0,B) (q,pB) (q,pB) (q,pB) (q,pB) (p,pB) (pq,pB) (pq,pB) (pq,pB) (pq,pB) (p,pB) (pq,pB) (pq,pB) (pq,pB) (pq,pB) (p,r) (p,pB) (pq,pB) (pq,pB) (pq,pB) (pq,pB) (p,pB) (pq,pB) (pq,pB) (pq,pB) (pq,pB) B*= {((p,r),(q,c),[F 1 (p,q),B(F 1 (p,r),c)]}

  25. X k (q,c) (0,c) (0,r) (r,c) (rq, qc) (rq, qc) (rq, qc) (rq, qc) (pr,pc) (pr,pc) (pr* k q, pq* k c) (p,r) (pr,pc) (pr,pc) X k ={((p,r),(q,c),[F k (F 1 (p,r),q),F k (F 1 (p,q),c)]}

  26. ( 0 , 0 ) ( 0 ,1 ) ( 0 , 3 ) ( 0 , 4 ) ( 0 , 2 ) ( 1 , 0 ) ( 1 , 1 ) ( 1 , 3 ) ( 1 , 4 ) ( 1 , 2 ) ( 0 , 2 ) ( 0 , 0 ) ( 0 , 1 ) ( 0 , 3 ) ( 0 , 4 ) ( 1 , 2 ) (1, 0 ) ( 1 , 1 ) ( 1 , 3 ) ( 1 , 4 ) ( 0 , 3 ) ( 0 , 4 ) ( 0 , 0 ) ( 0 , 2 ) ( 0 , 1 ) ( 1 , 3 ) (1, 4 ) ( 1 , 0 ) ( 1 , 2 ) ( 1 , 1 ) ( 0 , 4 ) ( 0 , 3 ) ( 0 , 2 ) ( 0 , 1 ) ( 0 , 0 ) ( 1 , 4 ) (1, 3 ) ( 1 , 2 ) ( 1 , 1 ) ( 1 , 0 ) ….. ( 0 , 1 ) ( 0 , 2 ) ( 0 , 4 ) ( 0 , 0 ) ( 0 , 3 ) ( 1 , 1 ) (1, 2 ) ( 1 , 4 ) ( 1 , 0 ) ( 1 , 3 ) ( 1 , 2 ) ( 1 , 0 ) ( 1 , 1 ) (1, 3 ) ( 1 , 4 ) ( 2 , 2 ) ( 2 , 0 ) ( 2 , 1 ) ( 2 , 3 ) ( 2 , 4 ) ( 1 , 3 ) ( 1 , 4 ) ( 1 , 0 ) ( 1 , 2 ) ( 1 , 1 ) ( 2 , 3 ) ( 2 , 4 ) ( 2 , 0 ) ( 2 , 2 ) ( 2 , 1 ) ( 1 , 4) ( 1 , 3 ) ( 1 , 2 ) ( 1 , 1 ) ( 1 , 0 ) ( 2 , 4 ) ( 2 , 3 ) ( 2 , 2 ) ( 2 , 1 ) ( 2 , 0 ) ( 1 , 1 ) ( 1 , 2 ) ( 1 , 4 ) ( 1 , 0 ) ( 1 , 3 ) ( 2 , 1) ( 2 , 2 ) ( 2 , 4 ) ( 2 , 0 ) ( 2 , 3 ) ( 1 , 0 ) ( 1 , 1 ) ( 1 , 3 ) ( 1 , 4 ) ( 1 , 2 ) ( 2 , 0 ) ( 2 , 1 ) ( 2 , 3 ) ( 2 , 4 ) ( 2 , 2 ) . . . . B*

  27. ( 0 , 0 ) ( 0 ,1 ) ( 0 , 2 ) ( 0 , 3 ) ( 0 , 4 ) ( 1 , 2 ) ( 1 , 3 ) ( 1 , 4 ) ( 1 , 0 ) ( 1 , 1 ) ( 2 , 0 ) ( 2 , 1 ) ( 2 , 2 ) ( 2 , 3 ) ( 2 , 4 ) ( 3 , 2 ) ( 3 , 3 ) ( 3 , 4 ) ( 3 , 0 ) ( 3 , 1 ) ( 4 , 0 ) ( 4 , 1 ) ( 4 , 2 ) ( 4 , 3 ) ( 4 , 4 ) ( 0 , 2 ) ( 0 , 3 ) ( 0 , 4 ) ( 0 , 0 ) ( 0 , 1 ) ( 1 , 0 ) ( 1 , 1 ) ( 1 , 2 ) ( 1 , 3 ) ( 1 , 4 ) ( 2 , 2 ) ( 2 , 3 ) ( 2 , 4 ) ( 2 , 0 ) ( 2 , 1 ) ….. ( 3 , 0 ) ( 3 , 1 ) ( 3 , 2 ) ( 3 , 3 ) ( 3 , 4 ) ( 4 , 2 ) ( 4 , 3 ) ( 4 , 4 ) ( 4 , 0 ) ( 4 , 1 ) ( 2 , 2 ) ( 2 , 3 ) ( 2 , 4 ) ( 2 , 0 ) ( 2 , 1 ) ( 3 , 4 ) ( 3 , 0 ) ( 3 , 1 ) ( 3 , 2 ) ( 3 , 3 ) ( 4 , 2 ) ( 4 , 3 ) ( 4 , 4 ) ( 4 , 0 ) ( 4 , 1 ) ( 0 , 4 ) ( 0 , 0 ) ( 0 , 1 ) ( 0 , 2 ) ( 0 , 3 ) ( 1 , 2) ( 1 , 3 ) ( 1 , 4 ) ( 1 , 0 ) ( 1 , 1 ) ( 2 , 4 ) ( 2 , 0 ) ( 2 , 1 ) ( 2 , 2 ) ( 2 , 3 ) ( 3 , 2 ) ( 3 , 3 ) ( 3 , 4 ) ( 3 , 0 ) ( 3 , 1 ) ( 4 , 4) ( 4 , 0 ) ( 4 , 1 ) ( 4 , 2 ) ( 4 , 3 ) ( 0 , 2 ) ( 0 , 3 ) ( 0 , 4 ) ( 0 , 0 ) ( 0 , 1 ) ( 1 , 4 ) ( 1 , 0 ) ( 1 , 1 ) ( 1 , 2 ) ( 1 , 3 ) . . . . X 2

  28. F 1 F 2 0 1 2 3 4 0 1 2 3 4 1 2 3 4 0 2 3 4 0 1 2 3 4 0 1 4 0 1 2 3 3 4 0 1 2 1 2 3 4 0 4 0 1 2 3 3 4 0 1 2 0 0 1 1 3 4 2 2 0 0 1 3 4 3 4 0 2 1 4 3 2 1 0 1 2 4 0 3 B

  29. Consequences ◼ This results improves Jenkin’s result ◼ First embed in B ◼ B has order m where 2 k =m > 2n ≥ 2 k-1 ◼ so 2 k-2 ≤ n <2 k-1 this gives us 2 k ≤ 4n <2 k+1 order of the embedding m 2 ≤ 16n 2 ◼ There are at least m-1 ≥ 2n orthogonal mate s. ◼ BONUS: EMBEDDING IS IDEMPOTENT

  30. Corollary ◼ Let L be a Latin square of order n with n ≥ 3 and n ≠6. Then L can be embedded in a Latin square B of order n 2 where B has at least two mutually orthogonal mates.

  31. Corollary ◼ Let L be a Latin square of order n with n ≥ 7 and n ≠ 10,18 or 22. Then L can be embedded in a Latin square B of order n 2 where B has at least 4 mutually orthogonal mates.

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