  The cyclic matching sequenceability of a simple graph $G$, denoted
$\mathrm{cms}(G)$, is the largest integer $s$ for which there exists a cyclic
ordering of the edges of $G$ so that every set of $s$ consecutive edges forms a
matching. In this paper we consider the minimum cyclic matching sequenceability
of $k$-regular graphs. We completely determine this for $2$-regular graphs, and
give bounds for $k \geq 3$.
