Are there conditions (e.g., bridgeless) under which a connected, regular, trivalent planar graph contains either a face whose boundary has length two or a face whose boundary has odd length? Since googling does not yield any results, nor does playing with the handshaking Lemma and the Euler formula, I suspect the general statement must be false. However, I cannot construct a counter example.
Many thanks!
Edit: I do not see how the first comment below applies here.