Bipartite regular graphs with fixed diameter

For given nonnegative integers k and D , we consider the problem of determining n,(k, D ) , the smallest number n for which there exists a k-regular bipartite graph on n vertices with diameter D . We solve the problem for all pairs ( k , D ) with D + 2 (mod 4) and D + 3 (mod 4), for all pairs ( k , D ) with k even or k prime and D + 3 (mod 4), for all pairs with D I 9 or k I 4, and for a few other pairs. In the remaining cases, we obtain lower and upper bounds for n,,(k, D ) . 0 7995 John Wdey & Sons, /nc.