Let G be a connected planar graph with 10 vertices. If the number of edges on each face is three, then the number of edges in G is ___.

GATE 2015 · Discrete Mathematics · Graph Planarity · medium

Answer: E = 24

1. Apply face-degree sum: Every face has exactly 3 edges, so 3 * F = 2E, giving F = (2/3) * E.
2. Substitute into Euler's formula: 10 - E + (2E/3) = 2 => 10 - E/3 = 2 => E/3 = 8 => E = 24.