From the paper https://web.math.princeton.edu/~pds/papers/girth6/paper.pdf, a cubic (every vertex is degree 3), planar graph has girth (smallest length face) strictly less than 6. It is also true that a cubic, bipartite (all faces even length) graph contains a square (or a 2-gon).
I am interested in planar graphs with exactly one degree 2 vertex, the rest degree 3. Is it true that such a graph must contain a face of length 1,2, 3 or 5? (In other words, a face of length less than or equal to 5 that is not a square.)