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.

Difficulty level: Easy task (using definitions and simple reasoning)
Reasoning task
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