Resolving sets of directed Cayley graphs for the direct product of cyclic groups

summary:A directed Cayley graph $C(\Gamma ,X)$ is specified by a group $\Gamma $ and an identity-free generating set $X$ for this group. Vertices of $C(\Gamma ,X)$ are elements of $\Gamma $ and there is a directed edge from the vertex $u$ to the vertex $v$ in $C(\Gamma ,X)$ if and only if there is a generator $x \in X$ such that $ux = v$. We study graphs $C(\Gamma ,X)$ for the direct product $Z_m \times Z_n$ of two cyclic groups $Z_m$ and $Z_n$, and the generating set $X = \{ (0,1), (1, 0), (2,0), \dots , (p,0) \}$. We present resolving sets which yield upper bounds on the metric dimension of these graphs for $p = 2$ and $3$