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Apr 11, 2015 at 13:14 comment added Gordon Royle No, the (4,5)-cage has 19 vertices so there's nothing smaller.
Apr 10, 2015 at 17:20 vote accept user62562
Apr 10, 2015 at 15:04 comment added Chris Godsil @gordonRoyle: I was thinking there might be examples on fewer than 19 vertices?
Apr 10, 2015 at 14:53 comment added Gordon Royle Actually for this size (19+ vertices), genreg will be much better. But as Chris says, there are zillions of these graphs, with 132 million already by 26 vertices. We'd normally expect most to be non-planar, so (again reiterating Chris) there's unlikely to be anything more special about them than what you started with (4-regular, girth 5).
Apr 10, 2015 at 12:17 history answered Chris Godsil CC BY-SA 3.0