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 ⌈n∕2⌉ paths. Let Gk 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 Gk, for every k≥3. Further, we prove that for every graph G in Gk on n vertices, there exists a partition of its edge set into n∕2 paths of lengths in {2k−1,2k,2k+1} and cycles of length 2k.