# Graph Theory Confusion

What is the most number of regions (including the outside region) can a planar graph with V vertexes partition split the plane into? (No self-loops or multiedges allowed)

Im stumped on this question i know outside is 1

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Try drawing some pictures. Start with 3 vertices and draw in as many edges as you can and see how many regions you have divided the plane into. Then try 4 vertices, then 5. Perhaps a pattern will emerge. –  MJD Mar 18 '12 at 18:29
Probably this will be necessary en.wikipedia.org/wiki/Euler_characteristic#Planar_graphs –  Graphth Mar 18 '12 at 20:06

Let $F$, $E$, $V$ be the number of faces, edges, and vertices, respectively. If some face has more than three sides that we can just make more faces by connecting non-adjacent vertices of that face. So we want every face to be a triangle. Therefore $E = 3F/2$. Plug this into $F-E+V=2$ and we get $\boxed{F = 2V-4}$ for $V\geq 3$.