An "open and planar quad mesh" (more description below) with 4 mesh faces has 9 vertices, the same mesh with 8 faces has 15 vertices (2 faces at every X-axis row, 4 faces at every Y-axis column)...etc.. What is the mathematical relationship between the number of mesh faces and its number of vertices?
The planar quad mesh in this case is a continuous collection of n rectangular faces sharing 4 edges (the 4 outer boundary edges of the mesh are open). The number of faces in the x direction can be different to the ones in the y direction.
I have looked at the Euler characteristic for a mesh that is homotopic to a sphere but do not know the relationship for the specific type of mesh described above.
A $n\times n$ square has $n^2$ quads and $(n+1)^2$ vertices.
A rectangular $m\times n$ mesh has $mn$ quads and $(m+1)(n+1)$ vertices.