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 P2k+1-decomposition. (C) 2015 Elsevier B.V. All rights reserved.
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