Completing partial transversals of Cayley tables of Abelian groups (Q820857)

Summary: In [J. Comb. Theory, Ser. A 103, No. 2, 349--362 (2003; Zbl 1025.05009)] \textit{M. Grüttmüller} proved that if \(n\geqslant 3\) is odd, then a partial transversal of the Cayley table of \(\mathbb{Z}_n\) with length \(2\) is completable to a transversal. Additionally, he conjectured that a partial transversal of the Cayley table of \(\mathbb{Z}_n\) with length \(k\) is completable to a transversal if and only if \(n\) is odd and either \(n \in \{k, k + 1\}\) or \(n \geqslant 3k - 1\). \textit{N. J. Cavenagh} et al. [Finite Fields Appl. 15, No. 3, 294--303 (2009; Zbl 1161.05016)] showed the conjecture is true when \(k = 3\) and \(n\) is prime. In this paper, we prove Grüttmüller's conjecture for \(k = 2\) and \(k = 3\) by establishing a more general result for Cayley tables of Abelian groups of odd order.