A couple of facts about the

Petersen graph (depicted somewhat poorly below):

It is non

hamiltonian. That is, there is no

cycle within it that contains every

vertex (also known as a

Hamilton cycle).
It is not a

planar graph. That is, there is no way to move the vertices and edges around (keeping them attatched!) so that none of the edges

intersect.

The vertices are designated by the shapes
made of #'s, and the edges are the lines between them.
#
__###__
__/ # \__
__/ | \__
__/ | \__
__/ # \__
__/ ### \__
__/ / # \ \__
# __/ | | \__ #
###,__ / \ __,###
# \__# | | #__/ #
\ ###------------/-------\------------### /
| #\_ | | _/# |
\ \_ / \ _/ /
| \_ | | _/ |
\ \__ / \ __/ /
| \_| |_/ |
\ /\_ _/\ /
| | \_ _/ | |
\ / \___/ \ /
| | _/ \_ | |
\ / _/ \_ \ /
| | _/ \_ | |
\ # _/ \_ # /
| ### ### |
\ _/ # # \_ /
| / \ |
\#/ \#/
###-----------------------------###
# #

I'm open to any suggestions--about how to improve this representation of the Petersen graph, about whether it's illustrative enough to even be worth it, about life in general...whatever.