ﻻ يوجد ملخص باللغة العربية
Let $P$ be a partial latin square of prime order $p>7$ consisting of three cyclically generated transversals. Specifically, let $P$ be a partial latin square of the form: [ P={(i,c+i,s+i),(i,c+i,s+i),(i,c+i,s+i)mid 0 leq i< p} ] for some distinct $c,c,c$ and some distinct $s,s,s$. In this paper we show that any such $P$ completes to a latin square which is diagonally cyclic.
We introduce a notion of parity for transversals, and use it to show that in Latin squares of order $2 bmod 4$, the number of transversals is a multiple of 4. We also demonstrate a number of relationships (mostly congruences modulo 4) involving $E_1,
An array is row-Latin if no symbol is repeated within any row. An array is Latin if it and its transpose are both row-Latin. A transversal in an $ntimes n$ array is a selection of $n$ different symbols from different rows and different columns. We pr
In combinatorics, a latin square is a $ntimes n$ matrix filled with n different symbols, each occurring exactly once in each row and exactly once in each column. Associated to each latin square, we can define a simple graph called a latin square grap
We prove a conjecture by Garbe et al. [arXiv:2010.07854] by showing that a Latin square is quasirandom if and only if the density of every 2x3 pattern is 1/720+o(1). This result is the best possible in the sense that 2x3 cannot be replaced with 2x2 or 1xN for any N.
We develop a limit theory of Latin squares, paralleling the recent limit theories of dense graphs and permutations. We introduce a notion of density, an appropriate version of the cut distance, and a space of limit objects - so-called Latinons. Key r