A minimum degree condition for the existence of S-path-systems in bipartite graphs

Let G be a graph, and S be a subset of V (G) with even cardinality. We denote the set of the endvertices of a path P by end(P). A path P is an S-path if |V (P)| >= 2 and V (P) boolean AND S = end(P). An l -S-path-system P is a set of vertex-disjoint S-paths such that S = UP is an element of P(V (P) boolean AND S) and |V (P )| <= l for each P is an element of P. In this paper, we show that if G is a bipartite graph with partite sets A and B with delta(G) >= max{|A|, |B|}/2, and if S is a subset of V (G) with even cardinality such that |A boolean AND S|-|B boolean AND S| <= 2|B\S| and |B boolean AND S| - |A boolean AND S| <= 2|A\S|, then, unless |A| = |B| is even and G is isomorphic to K|A|/2,|A|/2 boolean OR K|A|/2,|A|/2, G has a 6 -S-path-system P such that every path in P, possibly but one, has order two or three.