正负拉丁方与DINITZ猜想

Positive and Negative Latin Squares and Dinitz Conjecture

Ｄｉｎｉｔｚ猜想，ｎ×ｎ方格中，每一方格中各有ｎ个不同的元素，从每格中可选出一个元素，使各行各列均为相异代表系．ＪａｎｓｓｅｎＪＣＭ利用图的定向个数不等已证明了ｒ×ｎ（ｒ＜ｎ）时，Ｄｉｎｉｔｚ猜想成立．这里用代数方法把Ｄｉｎｉｔｚ猜想的解决与拉丁方联系了起来，并证明了，对于某ｎ，若所有ｎ阶拉丁方中正负个数不一样，则ｎ×ｎＤｉｎｉｔｚ猜想成立．于是当ｎ＝４时，Ｄｉｎｉｔｚ猜想解决．

The Dinitz Conjecture states that given an n×n array of n -sets, it is always possible to choose one element from each set, keeping the chosen elements distinct in every row and distinct in every column. By virtue of the number of orientations of the graph, Jeannette C. M. Jassen proved that the Dinitz Conjecture is true when the array is r×n(r<n). With the help of Latin Squares, this paper concludes that for any n,if the number of the positive Latin Squares is not equal to the number of the negative Latin Squares, then the n ×n Dinitz Conjecture is true. So the Dinitz Conjecture is true when n =4.