Theorem 1 (Euler's Formula) Let $G$ be a connected planar graph, and let $n, m$ and $f$ denote, respectively, the numbers of vertices, edges, and faces in a plane drawing of $G.$ Then $n - m + f = 2.$
Corollary 1 Let $G$ be a connected planar simple graph with $n$ vertices, where $n \geq 3$ and $m$ edges. Then $m \leq 3n - 6.$
Corollary 2 Let $G$ be a connected planar simple graph with $n$ vertices and $m$ edges, and no triangles. Then $m \leq 2n - 4.$
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