TITLE: 6-regular bipartite graphs with no 8-cycles
QUESTION [8 upvotes]: I'm looking for simple 6-regular bipartite graphs with no 8-cycles, as small as possible.  It doesn't matter if there are 4-cycles or 6-cycles, provided there are no 8-cycles.  Such graphs must exist since the girth can be arbitrarily high, but what smaller examples are there?  There are certainly none on less than 46 vertices.

REPLY [2 votes]: Hi Brendan.  I just happened upon this while searching for some related material, and I'm not sure if you're still interested in finding such a graph.  I'm relatively certain I can construct one on 7812 vertices, following a few old ideas used by Lazebnik, Ustimenko and me in a series of papers on extremal graphs (in the sense of Turan).  In fact the 7812-vertex graph I mention has girth 10, and its construction, as well as a proof that the girth is 10, should be easy to follow, and/or present.  Let me know if you're still interested.  I'll be looking back here every so often.  Or just drop me an email. Regards.