Path decompositions of regular graphs with prescribed girth

A P -decomposition of a graph G is a set of pairwise edge-disjoint paths of G with edges that cover the edge set of G. Kotzig (1957) proved that a 3-regular graph admits a P3-decomposition if and only if it contains a perfect matching, and also asked what are the necessary and sufficient conditions for an -regular graph to admit a P -decomposition, for odd . Let g, and m be positive integers with g ≥ 3. We prove that, (i) if is odd and m > 2 ( − 2)/(g − 2) , then every m -regular graph with girth at least g that contains an m-factor admits a P -decomposition; (ii) if m > ( − 2)/(g − 2) , then every 2m -regular graph with girth at least g admits a P -decomposition.