Row-Hamiltonian Latin squares and Falconer varieties

IF 1.5 1区数学 Q1 MATHEMATICS

Proceedings of the London Mathematical Society Pub Date : 2023-12-19 DOI:10.1112/plms.12575

Jack Allsop, Ian M. Wanless

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引用次数: 0

Abstract

A Latin square is a matrix of symbols such that each symbol occurs exactly once in each row and column. A Latin square

L $L$

is row-Hamiltonian if the permutation induced by each pair of distinct rows of

L $L$

is a full cycle permutation. Row-Hamiltonian Latin squares are equivalent to perfect 1-factorisations of complete bipartite graphs. For the first time, we exhibit a family of Latin squares that are row-Hamiltonian and also achieve precisely one of the related properties of being column-Hamiltonian or symbol-Hamiltonian. This family allows us to construct non-trivial, anti-associative, isotopically

L $L$

-closed loop varieties, solving an open problem posed by Falconer in 1970.

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行-汉密尔顿拉丁方阵和法尔科纳变体

拉丁方阵是一个符号矩阵，每个符号在每一行和每一列都正好出现一次。如果拉丁方阵 L$L$ 的每一对不同行所诱导的排列是全循环排列，那么拉丁方阵 L$L$ 就是行-哈密尔顿排列。行-哈密尔顿拉丁正方形等价于完整双方形图的完全 1 因式分解。我们首次展示了一个行-哈密顿拉丁平方族，它也恰好实现了列-哈密顿或符号-哈密顿的相关性质之一。通过这个族，我们可以构造出非对称、反联立、同素异形的 L$L$ 闭环品种，从而解决了法尔科纳在 1970 年提出的一个未决问题。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Proceedings of the London Mathematical Society 数学-数学

CiteScore

2.90

自引率

0.00%

发文量

审稿时长

6-12 weeks

期刊介绍： The Proceedings of the London Mathematical Society is the flagship journal of the LMS. It publishes articles of the highest quality and significance across a broad range of mathematics. There are no page length restrictions for submitted papers. The Proceedings has its own Editorial Board separate from that of the Journal, Bulletin and Transactions of the LMS.