On path decompositions of 2k-regular graphs

Tibor Gallai conjectured that the edge set of every connected graph $G$ on $n$ vertices can be partitioned into $\lceil n/2\rceil$ paths. Let $\mathcal{G}_{k}$ be the class of all $2k$-regular graphs of girth at least $2k-2$ that admit a pair of disjoint perfect matchings. In this work, we show that Gallai's conjecture holds in $\mathcal{G}_{k}$, for every $k \geq 3$. Further, we prove that for every graph $G$ in $\mathcal{G}_{k}$ on $n$ vertices, there exists a partition of its edge set into $n/2$ paths of lengths in $\{2k-1,2k,2k+1\}$.