I ran into a problem in Diestel's book Graph theory in planarity section:
Ancient greeks loved regular plane graphs whose faces were bounded by cycles of length $\ell$.
So, Diestel asks to prove that such graphs exist just for finitely many pairs $(d,\ell)$ of degree $d\geq 3$ and cycle of length $\ell$.
I tried to use Euler's characteristic with the number of edges $m=\frac{nd}{2}$ and obtain an upper bound but I just get that $$n\left(1-\frac{d}{2}\right)+f=2$$ where $f$ is the number of faces. But I don't know how to relate $f$ with $\ell$. Even I couldn't draw a graph of this style to get an idea how they look. Actually it also asks to prove that these graphs are unique up to topological isomorphism.
Any hint will be appreciate it :)
The equations we have our disposal are \begin{align} n-m+f=2,\tag{Euler's formula}\\ 2m=dn,\tag{Handshaking lemma}\\ 2m=f\ell.\tag{Handshaking lemma for dual graph} \end{align} If we substitute the expression $n=2m/d$ into $n$, and the expression $f=2 m/\ell$ into $f$, both in Euler's formula, we get $$ (2m/d)-m+(2 m/\ell)=2, $$ which becomes $$ \frac{2}{d}+\frac{2}{\ell} -1=\frac2{m}$$ Of course, $2/m>0$, so this implies $$ \frac{2}d+\frac2\ell> 1 $$ This inequality is enough to prove there are finitely many valid pairs $(d,\ell)$ with $d\ge 3$ and $\ell\ge 3$.