Related to girth of planar graphs 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.)
 A: You can prove half of that. One way to conclude something about the girth of a planar graph is via an inequality like this one: if the graph has $p$ vertices, $q$ edges, and girth $g$, then $q \le \frac{g(p−2)}{g−2}$.
For a cubic planar graph, we have $q = \frac32 p$, and we get an upper bound on $g$ that way. The upper bound is always less than $6$ (but approaches $6$ as $p \to \infty$).
If you have a single vertex of degree $2$, then $q = \frac32p - \frac12$, and the same argument still works. Solving for $g$, we get $g \le \frac{6p-2}{p+3} < \frac{6p}{p} = 6$. So there is always a face of length $1$, $2$, $3$, $4$, or $5$.
However, it's possible for all such faces to be squares. For example, take the skeleton graph of the truncated octahedron (which is $3$-regular) and then, to create a vertex of degree $2$, subdivide an edge that separates two hexagonal faces. You'll get a graph with $24$ degree-$3$ vertices and one degree-$2$ vertex, in which $6$ faces are squares, $6$ are hexagons, and $2$ are heptagons.
