Decomposition of (2k + 1)-regular graphs containing special spanning 2k-regular Cayley graphs into paths of length 2k + 1

Abstract

A Pℓ-decomposition of a graph G is a set of paths with ℓ 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 P2k+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 kth power of a spanning cycle; and for 5-regular graphs that contain special spanning 4-regular Cayley graphs.