Trades and defining sets in Latin squares and related combinatorial - PDF document

Trades and defining sets in Latin squares and related combinatorial arrays Nicholas J. Cavenagh University of Waikato July 9, 2019

Outline Synopsis Latin trades Defining sets History of lower bounds for sds ( n ) Equivalent ideas from computational learning theory Critical sets Arrays with frequency constraints Full Latin arrays (0,1)-arrays with constant row and column sum

Trades and defining sets Synopsis For a Latin square or other discrete mathematical structure, we can ask the following inter-related questions: • What is the minimum amount of information needed to define a structure uniquely? • What is the minimum amount of information possible to change one structure to another of the same order? • When can a partial structure be completed to one with specified parameters? In turn, these questions inform fundamental questions on enumeration and randomization of combinatorial structures. Nicholas J. Cavenagh | University of Waikato 3/33

Trades and defining sets MANY THINGS ARE CONNECTED BUT NOT EVERYTHING Nicholas J. Cavenagh | University of Waikato 4/33

Trades and defining sets Latin trades Let L 1 and L 2 be two Latin squares of the same order. 0 1 2 3 4 0 3 2 1 4 1 2 3 4 0 1 2 3 4 0 2 3 4 0 1 2 4 0 3 1 3 4 0 1 2 3 1 4 0 2 4 0 1 2 3 4 0 1 2 3 L 1 = L ( Z 5 ) L 2 Consider a Latin square to be a set of ordered (row, column, symbol) triples; for example L 1 = { ( r, c, r + c ) | r, c ∈ Z 5 } . Then the “difference” between L 1 and L 2 can be described by ( L 1 \ L 2 , L 2 \ L 1 ) . The distance between L 1 and L 2 is given by | L 1 \ L 2 | = 8 . Nicholas J. Cavenagh | University of Waikato 5/33

Trades and defining sets We say that L 1 \ L 2 is a Latin trade with disjoint mate L 2 \ L 1 . 1 3 3 1 3 4 0 4 0 3 4 0 1 1 4 0 L 1 \ L 2 L 2 \ L 1 Alternatively, a Latin trade T is a partially filled Latin square with disjoint mate T ′ such that: (i) T and T ′ occupy the same filled cells; (ii) T ∩ T ′ = ∅ ; (iii) Each row/column of T contains the same set of cells as the corresponding row/column of T ′ . Nicholas J. Cavenagh | University of Waikato 6/33

Trades and defining sets These two “definitions” of a Latin trade are not exactly the same!! 1 2 2 1 2 1 1 2 T T ′ Neither T nor T ′ embed in a Latin square of order 3 . Nicholas J. Cavenagh | University of Waikato 7/33

Trades and defining sets Defining sets A defining set for a Latin square L of order n is a subset D of L with unique completion to a Latin square of order n . 0 1 1 2 2 3 D It is conjectured that the smallest size for a defining set for a Latin square of order n , denoted by sds ( n ) , is ⌊ n 2 / 4 ⌋ . This is true for n ≤ 8 (Bean, 2005). Nicholas J. Cavenagh | University of Waikato 8/33

Trades and defining sets Lemma A subset D of a Latin square L is a defining set for L if and only if L \ D contains no Latin trade. 0 1 2 3 4 0 3 2 1 4 1 2 3 4 0 1 2 3 4 0 2 3 4 0 1 2 4 0 3 1 3 4 0 1 2 3 1 4 0 2 4 0 1 2 3 4 0 1 2 3 L 1 L 2 For example, the elements in blue are not a defining set. Nicholas J. Cavenagh | University of Waikato 9/33

Trades and defining sets Any two rows (or columns or symbols) of a Latin square form a trade. 0 1 2 3 4 1 2 3 4 0 1 2 3 4 0 0 1 2 3 4 2 3 4 0 1 2 3 4 0 1 3 4 0 1 2 3 4 0 1 2 4 0 1 2 3 4 0 1 2 3 L 1 L 2 Corollary The smallest defining set size sds ( n ) ≥ n − 1 . Nicholas J. Cavenagh | University of Waikato 10/33

Trades and defining sets History of lower bounds for sds ( n ) • sds ( n ) ≥ n + 1 , (Cooper, et al., 1994; Fu, et al., 1995) • sds ( n ) ≥ ⌊ (7 n − 3) / 6 ⌋ , (Fu, Fu, Rodger, 1997) • sds ( n ) ≥ ⌊ (4 n − 8) / 3 ⌋ , (Horak, Aldred, Fleischner, 2002) • sds ( n ) ≥ Ω( n !( n − 3)!) ∼ n log log n (Hermiston, 2018) • sds ( n ) ≥ n ⌊ (log n ) 1 / 3 / 2 ⌋ (C., 2007) • sds ( n ) ≥ Ω( n 3 / 2 ) (C., Ramadurai, 2017) • sds ( n ) ≥ n 2 / 10 4 (Hatami and Qian, 2018) Nicholas J. Cavenagh | University of Waikato 11/33

Trades and defining sets The prime Omega function Ω( n ) is the sum of the exponents in the prime factorization of the integer n . The following is a great table to show students.... n 2 3 4 5 6 7 8 9 10 ⌊ n 2 / 4 ⌋ 1 2 4 6 9 12 16 20 25 Ω( n !( n − 3)!) 1 2 4 6 9 12 16 20 23 Nicholas J. Cavenagh | University of Waikato 12/33

Trades and defining sets sds ( n ) ≥ Ω( n 3 / 2 ) (C., Ramadurai, 2017) This result is related to the fact that every Latin square has a trade of size O ( √ n ) . If p is prime, L ( Z p ) has no trade smaller than e log p + 2 (Drápal). It is conjectured this is best possible. Nicholas J. Cavenagh | University of Waikato 13/33

Trades and defining sets Consider the operation table for the integers as an infinite Latin square L ( Z ) := { ( a, b, a + b ) | a, b, ∈ Z } . Lemma The Latin square L ( Z ) has no finite Latin trades. Suppose for the sake of contradiction, there is a finite Latin trade T in L ( Z ) . Let k be the largest integer that appears in T as a symbol. Let i be the smallest integer such that ( i, i − k, k ) ∈ T . Then ( j, i − k, k ) ∈ T ′ for some j � = i . In fact j < i ; otherwise ( j, i − k, j + i − k > k ) ∈ T . Therefore ( j, j − k, k ) ∈ T , contradicting the minimality of i . Nicholas J. Cavenagh | University of Waikato 14/33

Trades and defining sets Corollary The following partial Latin square has no Latin trades. 0 1 2 0 1 2 3 0 1 2 3 4 1 2 3 4 2 3 4 Corollary The Latin square L ( Z n ) has a defining set of size ⌊ n 2 / 4 ⌋ . Corollary sds ( n ) ≤ ⌊ n 2 / 4 ⌋ . Nicholas J. Cavenagh | University of Waikato 15/33

Trades and defining sets The story behind sds ( n ) ≥ n 2 / 10 4 (Hatami and Qian, 2018) Theorem A Latin square of order n is equivalent to a partition of the edges of the complete tripartite graph K ( n, n, n ) into triangles. Label one partite set with the rows, one partite set with the columns and the remaining partite set with the symbols. Each (row, column, symbol) triple in the Latin square corresponds to a triangle in the graph. Nicholas J. Cavenagh | University of Waikato 16/33

Trades and defining sets The story behind sds ( n ) ≥ n 2 / 10 4 (Hatami and Qian, 2018) Theorem (Barber, Kühn, Lo, Osthus and Taylor 2016; Dukes 2015) Let γ > 0 and n > n 0 ( γ ) . Every balanced and locally balanced 3-partite graph on 3 n vertices with minimum degree at least (101 / 52+ γ ) n , admits a triangle decomposition. Corollary (BKLOT, 2016) Let P be a partial Latin square of order n > n 0 such that every row, column, and symbol is used at most 0 . 0288 n times. Then P can be completed to a Latin square. Nicholas J. Cavenagh | University of Waikato 17/33

Trades and defining sets Proof that sds ( n ) ≥ n 2 / 10 4 (Hatami and Qian, 2018): Let P be a partial Latin square of size at most n 2 / 10 4 . It suffices to show that if P completes to a Latin square L , P completes to a Latin square L ′ � = L (and thus is not a defining set. Let R , C and S be the rows, columns and symbols which are “crowded” (have more than 0 . 012 n entries). Fill all the empty cells in R , C and S by elements of L ; obtain a partial Latin square P ′ such that m rows, columns and symbols are all filled for some m ≤ 0 . 0084 n . Add an entry which is not in L . Create corresponding tripartite graph for the complement of P ′ , deleting vertices of degree 0 . It can be shown that the minimum degree of such a graph is less than 101 / 52( n − m ) . Nicholas J. Cavenagh | University of Waikato 18/33

Trades and defining sets Conjecture (Daykin and Häggkvist, 1984) Every balanced and locally balanced 3-partite graph on 3 n ver- tices with minimum degree at least 3 n/ 2 , ad- mits a triangle decomposition. Wanless (2002): The above is the best possible bound. For example: 1 0 0 1 0 1 1 0 2 3 3 2 2 The above conjecture appears not to be enough to prove that sds ( n ) ≥ ⌊ n 2 / 4 ⌋ . Nicholas J. Cavenagh | University of Waikato 19/33

Trades and defining sets Conjecture (Nash-Williams, 1970) Every sufficiently large even graph G with minimum degree 3 n/ 4 (and number of edges divisible by 3 ) has a triangle decomposition. Best-known upper bound for the above: Minimum degree 0 . 956 n (BKLO, 2015). Nicholas J. Cavenagh | University of Waikato 20/33

Trades and defining sets Equivalent ideas from computational learning theory (Hatami and Qian, 2018) expressed their results also using terminology from computational learning theory. teaching set (witness set, discriminant, specifying set)= defining set Nicholas J. Cavenagh | University of Waikato 21/33

Trades and defining sets Critical sets A critical set for a Latin square L is a minimal defining set for L . The size of the smallest (respectively, largest) critical set in L is denoted by sds ( L ) (respectively, lcs ( L ) ). Let L n be the set of Latin squares of order n . Then: sds ( n ) = min { sds ( L ) | L ∈ L n } = sds ( n ) , lcs ( n ) = max { lcs ( L ) | L ∈ L n } . We also define a type of infimum and supremum, respec- tively: inf ( n ) = max { sds ( L ) | L ∈ L n } ≤ lcs ( n ) , sup ( n ) = min { lcs ( L ) | L ∈ L n } ≥ sds ( n ) . Nicholas J. Cavenagh | University of Waikato 22/33