Constructions of Large Graphs on Surfaces

Abstract

We consider the degree/diameter problem for graphs embedded in a surface, namely, given a surface Σ and integers Δ and k, determine the maximum order N(Δ,k,Σ) of a graph embeddable in Σ with maximum degree Δ and diameter k. We introduce a number of constructions which produce many new largest known planar and toroidal graphs. We record all these graphs in the available tables of largest known graphs. Given a surface Σ of Euler genus g and an odd diameter k, the current best asymptotic lower bound for N(Δ,k,Σ) is given by

$$\sqrt{\frac{3}{8}}g \Delta^{\lfloor k/2 \rfloor}.$$

Our constructions produce new graphs of order

$$\left\{\begin{array}{ll}6 \Delta^{\lfloor k/2 \rfloor} \qquad \qquad \qquad \qquad {\rm if \Sigma\;is\;the\;Klein\;bottle} \\ \left(\frac{7}{2} + \sqrt{6g + \frac{1}{4}}\right) \Delta^{\lfloor k/2 \rfloor} \quad {\rm otherwise},\end{array}\right.$$

thus improving the former value.