Czechoslovak Mathematical Journal, Vol. 69, No. 3, pp. 621-636, 2019

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

Demelash Ashagrie Mengesha, Tomáš Vetrík

Received March 18, 2017.   Published online May 17, 2019.

Abstract:  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 $3$.
Keywords:  metric dimension; resolving set; Cayley graph; direct product; cyclic group
Classification MSC:  05C25, 05C12
DOI:  10.21136/CMJ.2019.0127-17

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Affiliations:   Demelash Ashagrie Mengesha, Tomáš Vetrík, Department of Mathematics and Applied Mathematics, University of the Free State, P. O. Box 339, Bloemfontein, 9300, Free State, South Africa, e-mail:,

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