Number of vertices of a planar graph
Task number: 3918
If a planar graph has a total of \( 12 \) faces, each is a pentagon and each vertex has degree \( 3 \), how many vertices does it have?
Solution
We determine the number of edges in two ways:
\(|E_G|=12{\cdot}5\cdot\frac12\), because \( G \) has 12 faces, each face has 5 edges and each edge separates two faces.
\(|E_G|=|V_G|\cdot3\cdot\frac12\) because each vertex has degree 3 and each edge has two ends.
Hence \(|V_G|=\frac{12{\cdot} 5}3=20\).
Answer
The graph has \( 20 \) vertices, it is the dodecahedron graph.
