Consider a planar graph with 2n vertices which has two faces that are n-gons, and all remaining faces are triangles. How many edges does this graph have?
2026-03-30 05:32:12.1774848732
Consider a planar graph with 2n vertices which has two faces that are n-gons, and all remaining faces are triangles.
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Draw two concentric $n-$gons with the inner one clocked slightly so that its vertices are between the vertices of the outer one. You have accounted for all $2n$ vertices, so you can't have any more. Now draw the band of obvious triangles between the $n-$gons. Each triangle has a base on one of the $n-$gons and two sides that it shares with other triangles, so there are $2n$ triangles. There are $n$ edges in each $n-$gon and one between each pair of neighboring triangles, for $4n$ in total. A version of the figure with $n=8$ is below.
If the question setter has been fair, s/he has promised you that the number of edges is the same for all configurations that meet the requirement, so you are done.