Degree sum conditions for path-factors with specified end vertices in bipartite graphs

Let GG be a graph, and let SS be a subset of the vertex set of GG. We denote the set of the end vertices of a path PP by end(P)end(P). A path PP is an SS-path if |V(P)|≥2|V(P)|≥2 and V(P)∩S=end(P)V(P)∩S=end(P). An SS-path-system is a graph HH such that HH contains all vertices of SS and every component of HH is an SS-path. In this paper, we give a sharp degree sum condition for a bipartite graph to have a spanning SS-path-system.