The degree-diameter problem in maximal bipartite planar graphs

Abstract

The (A ,D) (degree/diameter) problem consists of finding the largest possible number of vertices n among all the graphs with maximum degree and diameter D. We consider the (A ,D) problem for maximal planar bipartite graphs, that are simple planar graphs in which every face is a quadrangle. We obtain that for the ( , 2) problem, the number of vertices is n = + 2; and for the ( , 3) problem, n = 3 -1 if is odd and n = 3 -2 if is even. Then, we study the general case ( A ,D) and obtain that an upper bound on n is approximately 3(2D+1)( -2)bD/2c, and another one is C( - 2)bD/2c if D and C is a sufficiently large constant. Our upper bounds improve for our kind of graphs the one given by Fellows, Hell and Seyffarth for general planar graphs. We also give a lower bound on n for maximal planar bipartite graphs, which is approximately ( - 2)k if D = 2k, and 3( - 3)k if D = 2k + 1, for and D sufficiently large in both cases.