Distances of group tables and latin squares via equilateral triangle dissections

Szabados, Michal

doi:10.1016/j.jcta.2013.10.005

Cited by 6 publications

(11 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We do this by showing the existence of orthogonal Latin trades in B p with size O(log p). In order to prove Theorem 5.1 we modify a construction given by Szabados [10] which proved the following. Since our proof is a modification of that given in [10] (which was in turn inspired by classic results on dissections of squares by Brooks, Smith, Stone and Tutte [1,13] and Trustum [12]) we borrow from the notation given in [10].…”

Section: Orthogonal Trades Via Latin Trades In B Pmentioning

confidence: 99%

See 1 more Smart Citation

Orthogonal Trades in Complete Sets of MOLS

Cavenagh

Donovan²,

Demirkale³

2017

Electron. J. Combin.

View full text Add to dashboard Cite

Let B p be the Latin square given by the addition table for the integers modulo an odd prime p. Here we consider the properties of Latin trades in B p which preserve orthogonality with one of the p − 1 MOLS given by the finite field construction. We show that for certain choices of the orthogonal mate, there is a lower bound logarithmic in p for the number of times each symbol occurs in such a trade, with an overall lower bound of (log p) 2 / log log p for the size of such a trade. Such trades imply the existence of orthomorphisms of the cyclic group 1 which differ from a linear orthomorphism by a small amount. We also show that any transversal in B p hits the main diagonal either p or at most p−log 2 p−1 times. Finally, if p ≡ 1 mod 6 we show the existence of Latin square containing a 2 × 2 subsquare which is orthogonal to B p .

show abstract

Section: Orthogonal Trades Via Latin Trades In B Pmentioning

confidence: 99%

“…T := {(0, 0, 0), (0, 5, 5), (5, 0, 5), (5,3,8), (8,0,8), (5,5,10), (7, 3, 10), (7,4,11), (8,3,11), (7,5,12), (8,4,12), (8,5, 0)}. T ′ := {(0, 0, 5), (0, 5, 0), (5,0,8), (5,3,10), (8, 0, 0), (5, 5, 5), (7, 3, 11), (7,4,12), (8,3,8), (7,5,10), (8,4,11), (8,5,12)}.…”

Section: Orthogonal Trades Via Latin Trades In B Pmentioning

confidence: 99%

Orthogonal Trades in Complete Sets of MOLS

Cavenagh

Donovan²,

Demirkale³

2017

Electron. J. Combin.

View full text Add to dashboard Cite

show abstract

“…Early motivation [12] for their study arose from considering the differences between the operation tables of a finite group and a latin square of the same order, that is: what is the 'distance' between a group and a latin square? The study of the topological and geometric properties of latin trades has lead to significant progress towards understanding such differences, see for example [8,10,1,18,19], also see [6] for a survey of earlier results.…”

Section: Introductionmentioning

confidence: 99%

On which groups can arise as the canonical group of a spherical latin bitrade

Bonetta-Martin

McCourt

2017

Ars Math. Contemp.

View full text Add to dashboard Cite

We address a question of Cavenagh and Wanless asking: which finite abelian groups arise as the canonical group of a spherical latin bitrade? We prove the existence of an infinite family of finite abelian groups that do not arise as canonical groups of spherical latin bitrades. Using a connection between abelian sandpile groups of digraphs underlying directed Eulerian spherical embeddings, we go on to provide several, general, families of finite abelian groups that do arise as canonical groups. These families include:• any abelian group in which each component of the Smith Normal Form has composite order;• any abelian group with Smith Normal Form Z n p ⊕ k i=1 Z pa i , where 1 ≤ k, 2 ≤ a 1 , a 2 , . . . , a k , p and n ≤ 1 + 2 k i=1 (a i − 1); and • with one exception and three potential exceptions any abelian group of rank two.

show abstract

“…The upper bound in Conjecture cannot be decreased, since it is known that any Latin trade in

B_{n}

has size at least

e log p + 3

, where p is the least prime that divides n (). It was recently shown in that for each integer n ,

B_{n}

contains a Latin trade of size

5 {prefixlog}_{2} n

.…”

Section: Introductionmentioning

confidence: 99%

“…The upper bound in Conjecture 1.1 cannot be decreased, since it is known that any Latin trade in B n has size at least e log p + 3, where p is the least prime that divides n ( [4,9]). It was recently shown in [14] that for each integer n, B n contains a Latin trade of size 5 log 2 n. If Conjecture 1.1 above is true, it thus may be that the back circulant Latin square is the "loneliest" of all Latin squares; i.e. the Latin square with greatest minimum distance to any other Latin square.…”

Section: Introductionmentioning

confidence: 99%