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Euler Characteristic & Genus

Each Cayley Map has an Euler Characteristic, $\chi$ , such that $\chi=v-e+f$ , where $v$ is the number of vertices, $e$ is the number of edges, and $f$ is the number of faces. The $\chi$ of different Cayley Maps is the same if the Cayley Maps embed the same surface.

The genus, $g$ , is equal to the number of holes in the surface. Spheres have a genus of 0, one-holed toruses have a genus of 1, two-holed toruses have a genus of 2, etc. The genus of a graph can be found using the formula $\chi=2-2g$ .

According to Ringel, for any positive integer $n$ , the genus of the complete bipartite graph $K_{n,n}$ is $L(n,n)$ , where $L(n,n)$ if the ceiling of $\frac{(n-2)^2}{4}$ . Equations 2 and 3 show the formulas for even and odd integers, specifically.

Equation 2. For $n_e$ , where $n_e$ is an even integer, $L(n_e,n_e)=(\frac{n_e}{2})^2-n_e+1$ .

Equation 3. For $n_o$ , where $n_o$ is an odd integer, $L(n_o,n_o)=\frac{n_o^2+3}{4}-n_o+1$ .

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Euler Characteristic & Genus