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Constructions of Large Graphs on Surfaces

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 \[\sqrt{\frac{3}{8}g}Δ^{\lfloor k/2\rfloor}.\] Our constructions produce new graphs of order \[\begin{cases}6Δ^{\lfloor k/2\rfloor}& \text{if $Σ$ is the Klein bottle}\\ \(\frac{7}{2}+\sqrt{6g+\frac{1}{4}}\)Δ^{\lfloor k/2\rfloor}& \text{otherwise,}\end{cases}\] thus improving the former value by a factor of 4.