The Alon-Tarsi conjecture: A perspective on the main results

The Alon-Tarsi conjecture states that if n is even, then the sum of the signs of the Latin squares of order n is non-zero (Alon and Tarsi, 1992). The conjecture has been proven in the cases n = p + 1 (Drisko, 1997), and n = p 1 (Glynn, 2010), where p is an odd prime. This paper is intended to be a concise and largely self-contained account of these results, along with streamlined, and in some cases, original proofs that should be readily accessible to a mathematician with a background in combinatorics. We also discuss the relation between the Alon-Tarsi conjecture and Rota's basis conjecture (Huang and Rota, 1994), and present some related problems, such as Zappa's extension of the Alon-Tarsi conjecture (Zappa, 1997), and Drisko's proof of the extended conjecture for n = p (Drisko, 1998). (C) 2019 Elsevier B.V. All rights reserved.