Hamiltonian and long paths in bipartite graphs with connectivity

Let Gbe a graph,.(G) the order of G,.(G) the connectivity of Gand ka positive integer such that k <=(v(G) - 2)/2. Then Gis said to be k-extendableif it has a matching of size kand every matching of size kextends to a perfect matching of G. A Hamiltonian pathof a graph Gis a spanning path of G. A bipartite graph Gwith vertex sets V-1 and V-2 is defined to be Hamiltonian-laceableif such that | V-1| =|V-2| and for every pair of vertices p. V-1 and q epsilon V-2, there exists a Hamiltonian path in Gwith endpoints p and q, or | V-1| =| V-2| + 1and for every pair of vertices p, q epsilon V-1, p not equal q, there exists a Hamiltonian path in Gwith endpoints pand q. Let Gbe a bipartite graph with bipartition (X, Y). Define bn(G) to be a maximum integer such that 0 <= bn( G) < min{| X|, |Y|} and (1) for each non-empty subset S of X, if |S| <= | X| - bn(G), then |N(S)| >= |S| + bn(G), and if |X| - bn(G) < | S| <= | X|, then N( S) = Y; and (2) for each non-empty subset Sof Y, if |S| <= |Y| - bn(G), then |N(S)| >= |S| + bn(G), and if |Y| - bn(G) < | S| <= |Y|, then N(S) = X; and (3) bn(G) = 0if there is no non-negative integer satisfying (1) and (2). non-negative integer satisfying (1) and (2). Let Gbe a bipartite graph with bipartition ( X, Y) such that | X| =|Y| and bn(G) > 0. In this paper, we show that if v(G) <= 2 kappa(G) + 4bn( G) - 4, then Gis Hamiltonian-laceable; or if.(G) > 6bn(G) - 2, then for every pair of vertices x epsilon Xand y epsilon Y, there is an (x, y)-path Pin Gwith |V(P)| >= 6bn(G) - 2. We show some of its corollaries in k-extendable, bipartite graphs and a conjecture in k-extendable graphs.