Skolem Arrays and Skolem Labellings of Ladder Graphs

Abstract

We introduce Skolem arrays, which are two-dimensional analogues of Skolem sequences. Skolem arrays are ladders which admit a Skolem labelling in the sense of [2]. We prove that they exist exactly for those integers $n=0$ or $1(\mathrm{mod}4)$. In addition, we provide an exponential lower bound for the number of distinct Skolem arrays of a given order. Computational results are presented which give an exact count of the number of Skolem arrays up to order $16$.