I went to an undergrad's senior honors thesis presentation a few days ago. She was discussing crossing numbers and mentioned that complete graphs $K_n$ are nonplanar iff $n \geq 5$.
?Coincidentally? $A_n$ (the alternating group of even permutations) is a nonabelian simple group iff $n \geq 5$.
I know there are many connections between graph theory/finite geometry and group theory. It seems that many simple groups (especially sporadics) have roots in weird graphs and odd geometric objects. So this led me to wonder...
Is there a connection between $K_n$ being nonplanar and $A_n$ being nonabelian simple (when $n \geq 5$)?
...or maybe this is just a coincidence.
EDIT: I just posted this on Mathoverflow to see if I get any response there. https://mathoverflow.net/questions/150746/a-connection-between-nonplanar-complete-graphs-and-the-alternating-group