Path factors in bipartite graphs from size or spectral radius

Let G be a graph and let \(P_n\) be a path on n vertices. A spanning subgraph H of G is called a \(\{P_{3},P_{4},P_{5}\}\)-factor if every component of H is one of \(P_3,\, P_4\) and \(P_5\). In 1994, Wang (J Graph Theory 18(2):161–167, 1994) gave a sufficient and necessary condition to ensure that a bipartite graph contains a \(\{P_{3},P_{4},P_{5}\}\)-factor. In this paper, we use an equivalent form of Wang-type condition to establish two sufficient conditions to ensure that there exists a \(\{P_{3},P_{4},P_{5}\}\)-factor in a connected bipartite graph, in which one is based on the size, the other is based on the spectral radius of the bipartite graph.