\(P_{2p}\)-factorization of a complete bipartite graph (Q687148)

Given a graph \(G\), every spanning subgraph with all its components isomorphic to the \(2k\)-vertex path is called a \(P_{2k}\)-factor of \(G\), for a fixed integer \(k\geq 1\). It is shown that the complete bipartite graph \(K_{n,m}\) is the edge-disjoint union of \(P_{2k}\)-factors if and only if \(m=n\) and \(m\equiv 0\mod{k(2k-1)}\).