The degree/diameter problem in maximal planar bipartite graphs

Abstract

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 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 (∆, D) and obtain that an upper bound on n is approximately 3(2D + 1)(∆ − 2) ⌊D/2⌋ and another one is C(∆ − 2) ⌊D/2⌋ if ∆ ≥ 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 (∆ − 2) k if D = 2k, and 3(∆ − 3) k if D = 2k + 1, for ∆ and D sufficiently large in both cases.

New results on the degree/diameter problem of mixed Abelian Cayley graphs

“…Besides this general bound given above, researchers are also interested in some particular versions of the problem, namely when the graphs are restricted to a certain class, such as the class of bipartite graphs (which was studied by the authors [3]), planar graphs (see Fellows, Hell, and Seyffarth [6], and Tischenko [20]), maximal planar bipartite graphs (see Dalfó, Huemer, and Salas [4]), vertex-transitive graphs (see Machbeth, Šiagiová, Širáň, and Vetrík [12], and Šiagiová and Vetrík [18]), Cayley graphs ( [12,18] and Vetrík [21]), Cayley graphs of Abelian groups (Dougherty and Faber [5]), or circulant graphs (Wong and Coppersmith [22], and Monakhova [15]). In this paper, we are concerned with mixed Abelian Cayley graphs.…”

New Moore-Like Bounds and Some Optimal Families of Abelian Cayley Mixed Graphs

“…This approximation is good only for small values of z. For instance, Circ(n 2 ; {1, n, 1 2 n 2 }) approaches the upper bound by the factor 4 9 . The following result shows a better family of dense mixed graphs with r = 1 and z ≥ 2. .…”