TITLE: Looking for cubic, bipartite graphs with girth at least six and no cycles of length 8.
QUESTION [6 upvotes]: Aside from the Desargues graph, are there nice (at least vertex-transitive), small (say, less than 60 vertices), cubic, bipartite graphs with girth at least 6 and no 8-cycles? (or, even better, no cycles of length congruent to 0 mod 4.) 
Bonus: any idea how to use Mathematica to pull these out of its list of graphs it's got data for in Mathematica 7?

REPLY [5 votes]: Desargues graph, also known as the generalized Petersen graph $G(10,3)$ has girth 6 and also contains cycles of length 8. There exist three 10-cages, smallest cubic graphs of girth 10. They have 70 vertices and none of them is vertex-transitive. For more information about the motivation
see here. 
Therefore it is quite surprising that Gordon's example is so small. By the way, it is also arc-transitive. One may consider it as a Levi graph of a self-dual, flag-transitive 3-configuration of 19 points and 19 lines. Configuration contains triangles but has no quadrangles.
Here is a geometric realization of this graph.
Configuration with triangles but no quadrangles

A natural question is, whether this is the smallest graph with required properties. 
Another related question is, whether there exist smaller bipartite graphs if the vertex-transitivity condition is dropped.