Decompositions of triangle-free 5-regular graphs into paths of length five

Abstract

A $P_k$-decomposition of a graph  $G$ is a set of edge-disjoint paths with $k$  edges that cover the edge set of  $G$. Kotzig (1957) proved that a 3-regular graph admits a $P_3$-decomposition if and only if it contains a perfect matching. Kotzig also asked what are the necessary and sufficient conditions for a $(2k+1)$-regular graph to admit a decomposition into paths with $2k+1$  edges. We partially answer this question for the case $k=2$ by proving that the existence of a perfect matching is sufficient for a triangle-free 5-regular graph to admit a $P_5$-decomposition. This result contributes positively to the conjecture of Favaron et al. (2010) that states that every  $(2k+1)$-regular graph with a perfect matching admits a $P_{2k+1}$-decomposition.