I'm working on a problem in planar graphs, and I came across an interesting problem related to a graph of a soccer ball. Consider a soccer ball made entirely of hexagons and pentagons. Each "vertex" of the soccer ball - the points where three shapes meet, has a degree of three. I want to prove that the number of hexagons and pentagons are constant for any soccer ball.
To some extent, we can imagine the soccer ball as a graph, where each vertex $v_i$ is a point where three shapes meet, and the edges are lines where two shapes meet. Let us consider the number of hexagons $H$ and the number of pentagons $P$ in the graph. I was given the suggestions to consider the bipartition of pentagons $P$ and hexagons $H$, where there is an edge between pentagon $p_i$ and hexagon $h_i$ if they share an edge on the soccer ball. I'm trying to figure out some properties of this bipartition that may give me some information as to a formula between the number of pentagons and the number of hexagons. Given that this graph is likely planar, I would also like to figure out functions of the number of vertices and the number of edges in terms of $P$ and $H$. Any suggestions on this problem?