Given that a graph has 12 vertices and every region has four boundary edges, I need to find its number of edges, regions, and then prove if it is planar.
2026-02-22 19:07:20.1771787240
Proving a graph is planar given vertices and boundary edges
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Let $v=12$ be the number of vertices, let $e$ be the number of edges and let $f$ be the number of regions.
The graph is planar so by Euler we have $v-e+f=2$. Each region has $4$ edges (this will double count the edges) so $4f=2e$ (or $2f=e$) so ...