The Degree/Diameter Problem in Maximal Planar Bipartite graphs

The (∆, 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 (∆, D) problem for maximal planar bipartite graphs, that are simple planar graphs in which every face is a quadrangle. We prove 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 (∆, 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) if D = 2k (k ≥ 1), and 3(∆− 3) if D = 2k + 1 (k ≥ 4), for ∆ and D sufficiently large in both cases.