Why is do no complete graphs have genus 7? My teacher mentioned this in class, but he rushed his explanation due to time restraints. Why is it that complete graphs ($K_n$) never have genus $7$?
 A: It is a theorem of Ringel and Youngs that for all $n \geq 3$, the genus of the complete graph $K_n$ -- i.e., the minimal genus of a compact orientable surface into which $K_n$ embeds -- is $\lceil \frac{(n-3)(n-4)}{12} \rceil$.  Granting this result, what you ask about is very straightforward: the given function is weakly increasing.  For $n = 12$ it takes the value $6$.  For $n = 13$ it takes the value $8$.  Thus it never takes the value $7$ (the first of infinitely many values that it skips).
Not being a graph theorist, I confess that I don't know the proof of the Ringel-Youngs Theorem, and the paper was not freely available to me on the internet.  For all I know it may be possible to avoid this result and give an argument more specific to your precise question...but I will guess that it will probably be more fruitful to learn the proof of the general result.  

Ringel, G. and Youngs, J. W. T. "Solution of the Heawood Map-Coloring Problem." Proc. Nat. Acad. Sci. USA 60, 438-445, 1968. 
