The degree/diameter problem in maximal planar bipartite graphs

Abstract

The (Δ,D) (degree/diameter) problem consists of finding the largest possible number of vertices n among all the graphs with maximum degree Δ and diameter D. We consider the (Δ,D) problem for maximal planar bipartite graphs, that are simple planar graphs in which every face is a quadrangle. We obtain that for the (Δ,2) problem, the number of vertices is n=Δ+2; and for the (Δ,3) problem, n=3Δ−1 if Δ is odd and n=3Δ−2 if Δ is even. Then, we study the general case (Δ,D) and obtain that an upper bound on n is approximately 3(2D+1)(Δ−2)⌊D/2⌋ and another one is C(Δ−2)⌊D/2⌋ if Δ≥D and C is a sufficiently large constant. Our upper bound 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 (Δ−2)k if D=2k, and 3(Δ−3)k if D=2k+1, for Δ and D sufficiently large in both cases.