Constructions of Large Graphs on Surfaces

TL;DR

This paper explores the degree/diameter problem for graphs embedded in surfaces, introducing new constructions that significantly improve the known bounds for maximum graph order on various surfaces.

Contribution

It presents novel constructions of large graphs on surfaces like the Klein bottle and others, improving existing bounds for the maximum order of such graphs.

Findings

New constructions produce larger graphs than previously known.

Improved asymptotic lower bounds for graphs on surfaces.

Enhanced tables of largest known graphs for various surfaces.

Abstract

We consider the degree/diameter problem for graphs embedded in a surface, namely, given a surface $\Sigma$ and integers $\Delta$ and $k$, determine the maximum order $N(\Delta,k,\Sigma)$ of a graph embeddable in $\Sigma$ with maximum degree $\Delta$ 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 $\Sigma$ of Euler genus $g$ and an odd diameter $k$, the current best asymptotic lower bound for $N(\Delta,k,\Sigma)$ is given by \[\sqrt{\frac{3}{8}g}\Delta^{\lfloor k/2\rfloor}.\] Our constructions produce new graphs of order \[\begin{cases}6\Delta^{\lfloor k/2\rfloor}& \text{if $\Sigma$ is the Klein bottle}\\ \(\frac{7}{2}+\sqrt{6g+\frac{1}{4}}\)\Delta^{\lfloor k/2\rfloor}& \text{otherwise,}\end{cases}\] thus improving the former value by a factor of 4.