A graph G is almost regular or more precisely is a (d, d+ 1)-graph, if the degree of each vertex of G is either d or d + 1. Let d ≥ 2 be an integer, and let G be a connected bipartite (d, d + 1)-graph with partite sets X and Y such that |X| = |Y |. If the order of G is at most 4d + 4, then we show in this paper that G contains a perfect matching. Examples will demonstrate that the given bound o...