Completing partial transversals of Cayley tables of Abelian groups

Jaromy Kuhl¹ ; Donald McGinn¹ ; Michael William Schroeder²

¹University of West Florida
²Marshall University

The electronic journal of combinatorics, Tome 28 (2021) no. 3

Résumé

In 2003 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 $n \geqslant 3k - 1$. Cavenagh, Hämäläinen, and Nelson (in 2009) 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.

Détail

DOI : 10.37236/9386

Classification : 05D15, 05B15, 20K01
Mots-clés : Grüttmüller's conjecture