The bipartite regulation number br(G) of a bipartite graph G with maximum degree d is the minimum number of vertices required to add to G to construct a d-regular bipartite
by-n bipartite graph with m <~ n and n -m >~ d -1, then br(G) = n -m. If, however, n -m ~< d -2, then br(G) = n -m + 2/for some l satisfying 0 ~< l ~< d-(n-m). Conversely, if l, k and d (>2) are integers such that 0 <~ l <~ k and 2 <~ k <~ d, then there is an connected m-by-n bipartite graph G of maximum degree d for which br(G) = n -m + 2/, for some m and n with k = d -(n -m).
A well-known result of KOnig [9] states that every graph G of maximum degree d is an induced subgraph of a d-regular graph H. Erd6s and Kelly [5] determined the minimum number of vertices (the induced regulation number) which must be added to G to obtain such a supergraph H. This latter result was extended to digraphs by Beineke and Pippert [3].
The regulation number of a graph G is the minimum number of vertices which must be added to G to construct a d-regular supergraph H. In this case, G need not be an induced subgraph of H. Regulation numbers of graphs were introduced by Akiyama, Era and Harary [1], and were further studied by Akiyama and Harary [2] and Harary and Schrnidt . Analogous concepts for digraphs and multigraphs were introduced by Harary and Karabed [7] and Chartrand, Harary and Oellermann [4], respectively. Here we consider the bipartite regulation numbers.