I think I have some examples derived from the Peterson graph -- in fact, they are infinite "covers" of the Peterson graph. One example below, after twenty blank lines. All even numbers 2n are connected to 2n+1, 2n+2, and 2n-2. All odd numbers 2n+1 are connected to 2n, 2n+5, and 2n-4. I am pretty sure this graph is cubic of girth 5. On Sat, May 16, 2020 at 2:58 PM Keith F. Lynch <kfl@keithlynch.net> wrote:

Find a connected graph of girth 5 (i.e. no loops of size 3 or 4) in which every integer is a vertex, every vertex has the same valence, and every vertex is part of a loop of size 5.

Yes, I have a solution, unless I'm confused. As usual, I'll give it in a week if I get no solutions or requests for more time.

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