TITLE: Request for examples of 4-regular, non-planar, girth at least 5 graphs
QUESTION [5 upvotes]: Edit: As David Eppstein points out (in his answer below) the assumption that the graph is non-planar is redundant. 
Thank you to everyone who answered/commented.

I have a problem about geometric embeddings of graphs for which the case I cannot prove is when the (simple, connected) graph is 4-regular, non-planar and has girth at least 5.
I would like to get some intuition for such graphs - e.g. 
*small(est) examples, 
*do such graphs have any interesting special properties?
*I assume there are many when the number of vertices is large, 
*a book or paper that might help.
Apologies if this is too easy for math overflow, I'm not a graph theorist.

REPLY [9 votes]: If a planar graph has girth four or more, it can have at most $2n-4$ edges, but every 4-regular graph has exactly $2n$ edges, so every 4-regular graph with girth $\ge 4$ is nonplanar. That is, your requirement that the graph be nonplanar is redundant.
The projective plane of order 3 has 13 points, 13 lines, four points per line and four lines per point. Its Levi graph (a graph with 26 vertices, one for each point and one for each line, and with an edge for each point-line incidence) is bipartite with girth six. According to the link in the comment by user35593 it is the unique smallest 4-regular graph with this girth.
You can get bigger examples like this from other configurations with four points per line and four lines per point, such as the 256 points and 256 axis-parallel lines of a $4\times 4\times 4\times 4$ hypercube.