A degree sum condition for the existence of an [formula omitted]-path-system in a bipartite graph

Abstract

Let G be a graph, and S be a subset of V(G) with even order. We denote the set of the end vertices of a path P by end(P). A path P is called an S-path if |V(P)|≥2 and V(P)∩S=end(P). An l-S-path-system P is a set of vertex-disjoint S-paths such that S=⋃P∈P(V(P)∩S) and |V(P)|≤l for any P∈P. In this paper, we give a sharp degree sum condition for a bipartite graph to have an l-S-path-system such that l is small.