Cubic graphs (graphs in which every vertex has valency 3) cannot be trees, so they contain a lot of cycles. Some of these cycles have length divisible by 3 (e.g. triangles, hexagons, nonagons etc).
In a generic cubic graph some of these cycles will have chords (edges connecting non-adjacent vertices in the cycle) and some won't. My interest is in graph in which every cycle of length divisible by 3 has a chord. In particular such a graph can contain no triangles as these never have chords.
I constructed some examples, all of which were bipartite. However I don't see any natural reason why such a graph should necessarily be bipartite so my question is:
Does anyone know an example of a non-bipartite cubic graph in which every cycle of length divisible by 3 has a chord?