Decomposing \(2k\)-regular graphs into paths of length \(k\) (Q2164990)

A decomposition of a graph is a set of subgraphs that partition its edge set. If all these subgraphs are isomorphic to \(H\), then it is called an \(H\)-decomposition. A path decomposition \(\mathcal{D}\) is balanced if each vertex is a terminal of exactly two paths of \(\mathcal{D}\). In this paper, the authors verify that a \(2k\)-regular graph \(G\) possesses a balanced \(P_k\)-decomposition if every cycle of length at most \(k\) is induced in \(G\), which settles partly \textit{M. Kouider} and \textit{Z. Lonc}'s conjecture: a \(2k\)-regular graph \(G\) has a balanced \(P_k\)-decomposition [Australas. J. Comb. 19, 261--274 (1999; Zbl 0938.05053)].