I ran into a Graph Problem.
Suppose G is A Planner Graph with 100 Vertices such that if connect each two Non-adjacent vertices, the resulting graph would be non-planner. what is the number of edges of G?
Thanks
2026-03-31 10:03:17.1774951397
Planner Combination Problem on Graph
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I assume that you are talking about a simple graph (no loops, no multiple edges). Then every region in the graph has degree at least $3$. However if there is a region of degree more than $3$ then your condition is not satisfied. So every region has degree $3$ exactly. Also, the graph is connected as otherwise your condition is not satisfied. Therefore we have $$r+v=e+2\ ,\quad 3r=2e\ ,\quad v=100$$ from which you can find $r=196$ and $e=294$.