The cyclic matching sequenceability of regular graphs

The cyclic matching sequenceability of a simple graph $G$, denoted $\mathrm{c}\mathrm{m}\mathrm{s}(G)$, 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 $k\ge 3$.