How not to prove the Alon-Tarsi conjecture (Q2880835)

A Latin square of order \(n\) has sign \(-1\) if an odd number of its rows and columns (combined) are odd permutations of the tuple \((0,1,\dots,n-1)\), otherwise it's sign is \(+1\). Let \(L_n^O\) and \(L_n^E\) denote the number of sign \(-1\) and sign \(+1\) Latin squares of order \(n\), respectively. When \(n\) is odd it is known that \(L_n^O=L_n^E\), but when \(n\) is even the value of \(L_n^O-L_n^E\) is unknown in general. The Alon-Tarsi Conjecture [\textit{N.\ Alon} and \textit{M.\ Tarsi}, ``Colorings and orientations of graphs,'' Combinatorica 12, No. 2, 125--134 (1992; Zbl 0756.05049)] asserts that \(L_n^O\neq L_n^E\) when \(n\) is even. More than just a curiosity, this conjecture has connections to a set-theoretical conjecture of Dinitz (later proved in [\textit{F.\ Galvin}, ``The list chromatic index of a bipartite multigraph,'' J. Comb. Theory, Ser. B 63, No.\,1, 153--158 (1995; Zbl 0826.05026)]), and Rota's basis conjecture [\textit{R.\ Huang} and \textit{G.-C.\ Rota}, ``On the relations of various conjectures on Latin squares and straightening coefficients,'' Discrete Math. 128, No.\,1-3, 225--236 (1994; Zbl 0797.05019)]. Drisko has shown that if \(p\) is an odd prime then \(L_{p+1}^E-L_{p+1}^O\equiv (-1)^{(p+1)/2}p^2 \pmod{p^3}\), thus establishing the Alon-Tarsi conjecture in some special cases see [\textit{A.\ A.\ Drisko}, ``On the number of even and odd Latin squares of order \(p+1\),'' Adv. Math. 128, No. 1, 20--35 (1997; Zbl 0885.05034)]). In the present article the authors show that \(L_{n+1}^E \not\equiv L_{n+1}^O \pmod{t^3}\) only if \(t=n\) and \(n\) is an odd prime, thereby establishing limitations of Drisko's method. In addition the authors conjecture that \(L_n^E\) and \(L_n^O\) are asymptotically equal, and they introduce a generalization of the Alon-Tarsi conjecture. Overall this paper is well written, informative, entertaining, and largely self contained.