The Degree/Diameter Problem in Maximal Planar Bipartite graphs

Cristina Dalfó,Julián Salas,Clemens Huemer

doi:10.37236/4468

Abstract

The $(\Delta,D)$ (degree/diameter) problem consists of finding the largest possible number of vertices $n$ among all the graphs with maximum degree $\Delta$ and diameter $D$. We consider the $(\Delta,D)$ problem for maximal planar bipartite graphs, that is, simple planar graphs in which every face is a quadrangle. We obtain that for the $(\Delta,2)$ problem, the number of vertices is $n=\Delta+2$; and for the $(\Delta,3)$ problem, $n= 3\Delta-1$ if $\Delta$ is odd and $n= 3\Delta-2$ if $\Delta$ is even. Then, we prove that, for the general case of the $(\Delta,D)$ problem, an upper bound on $n$ is approximately $3(2D+1)(\Delta-2)^{\lfloor D/2\rfloor}$, and another one is $C(\Delta-2)^{\lfloor D/2\rfloor}$ if $\Delta\geq D$ 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 $(\Delta-2)^{k}$ if $D=2k$, and $3(\Delta-3)^k$ if $D=2k+1$, for $\Delta$ and $D$ sufficiently large in both cases.

Highlights

We consider simple graphs G = G(V, E) that are bipartite, planar and with the maximum possible number of edges
∗Research supported by the Ministry of Education and Science, Spain, and the European Regional Development Fund under project MTM2014-60127-P, and by the Catalan Research Council under project 2014SGR1147
From the Euler characteristic |V | − |E| + |F | = 2, which relates the numbers of vertices n = |V |, edges |E| and faces |F | in a planar embedding of G, and the fact that each face is incident to four edges, one obtains the well-known relations |E| = 2n − 4 and |F | = n − 2

Summary

Introduction

Consider a maximal planar bipartite graph G with diameter D = 3, maximum degree ∆ and maximum number of vertices n, n=. If G is a maximal planar bipartite graph with diameter D = 3, maximum degree ∆ = 4 and n vertices, n 10 Let G be a maximal planar bipartite graph with diameter D = 3, maximum degree ∆ 5 and n vertices. Case (1): w = 3: First consider the subcase when all three white faces share a common vertex v such that all the white vertices of degree two in G \ G are incident to v, see Figure 3 (left). This concludes the proofs of this lemma and the one of Theorem 2

An upper bound

A lower bound