  A $P_\ell$-decomposition of a graph $G$ is a set of paths with $\ell$ edges
in $G$ that cover the edge set of $G$. Favaron, Genest, and Kouider (2010)
conjectured that every $(2k+1)$-regular graph that contains a perfect matching
admits a $P_{2k+1}$-decomposition. They also verified this conjecture for
$5$-regular graphs without cycles of length $4$. In 2015, Botler, Mota, and
Wakabayashi verified this conjecture for $5$-regular graphs without triangles.
In this paper, we verify it for $(2k+1)$-regular graphs that contain the $k$th
power of a spanning cycle; and for $5$-regular graphs that contain special
spanning $4$-regular Cayley graphs.
