\(P_ 3\)-factorization of complete bipartite graphs
Let \(P_ 3\) be a path of 3 points and \(K_{m,n}\) be a complete bipartite graph with partite sets \(V_ 1\) and \(V_ 2\), where \(| V_ 1| =m\) and \(| V_ 2| =n.\) A spanning subgraph F of \(K_{m,n}\) is called a \(P_ 3\)-factor if each component of F is isomorphic to \(P_ 3\). If \(K_{m,n}\) is expressed as a line-disjoint sum of \(P_ 3\)-factors, then this sum is called a \(P_ 3\)-factorization of \(K_{m,n}.\) In this paper, it is shown that a necessary and sufficient condition for the existence of a \(P_ 3\)-factorization of \(K_{m,n}\) is (i) \(m+n\equiv 0\) (mod 3), (ii) \(m\leq 2n\), (iii) \(n\leq 2m\) and (iv) \(3mn/2(m+n)\) is an integer.
scientific article; zbMATH DE number 4091553