\(P_{4k-1}\)-factorization of complete bipartite graphs (Q2574687)

The authors use the term \(P_{4k-1}\)-factorization of \(K_{m,n}\) in the following sense. Let \(m+n=q(4k-1)\) and \(F\) be the vertex-disjoint union of \(q\) copies of the path \(P_{4k-1}\). A \(P_{4k-1}\)-factorization is then a set of edge-disjoint copies of \(F=qP_{4k-1}\) which partition the edge set of \(K_{m,n}\). As necessary and sufficient conditions for the existence of a \(P_{4k-1}\)-factorization of \(K_{m,n}\) are given: \((1)~(2k-1)m\leq 2kn,\) \((2)~ (2k-1)n\leq 2km,\) \((3)~m+n\equiv 0\pmod {4k-1},\) \((4)~2(2k-1)(m+n)| (4k-1)mn\). The result complements previously known results by \textit{H. Wang} [Discrete Math. 120, 307--308 (1993; Zbl 0781.05040)] on \(P_{2k}\)-factorization and by \textit{K. Ushio} [Discrete Math. 116, 299--311 (1993; Zbl 0783.05034)] on \(P_{3}\)-factorization of \(K_{m,n}\).