Perfect toroidal graphs

Question

A toroidal graph is perfect if every vertex has same degree $d$ and every face has the same number of edges $k$. I am asked to come up with possible values of $d$ and $k$ and to construct an innite series of examples of perfect toroidal graphs. But I am a little lost. I tried starting from the toroidal property ( Euler characteristic $= 0$) but it didn't lead me anywhere. The only relationship I came up with is $dV= 2E$ but I don't know how to approach$k$. Any hints/suggestions?

Answer 1

$V-E+F=0$ from the Euler characteristic. As you say, $dV = 2E$ because both count half-edges. But $kF$ also counts half-edges. So $2E/d - E + 2E/k = 0$ or $2k + 2d = dk$. Thus $d=\frac{2k}{k-2}$ and vice versa. Can you take it from here?

Answer 2

You know how to "make" a torus by taking a square and identifying opposite edges in pairs? Well, before you do the identification, pick a positive integer $n$, and draw in an $n\times n$ grid on the square. Now when you do the identification, that grid becomes a graph on the torus, where every vertex has degree 4, and every face has 4 edges. So, there's an infinite family.