  This article is about the graph genus of certain well studied graphs in
surface theory: the curve, pants and flip graphs. We study both the genus of
these graphs and the genus of their quotients by the mapping class group. The
full graphs, except for in some low complexity cases, all have infinite genus.
The curve graph once quotiented by the mapping class group has the genus of a
complete graph so its genus is well known by a theorem of Ringel and Youngs.
For the other two graphs we are able to identify the precise growth rate of the
graph genus in terms of the genus of the underlying surface. The lower bounds
are shown using probabilistic methods.
