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I just didn't understand the second part of the prove

"G has twice as many edges as vertices only "... what do I actually have to prove ?

I understand it like $n=2e$ , is it right ?

then isn't it's just similar to the degree equation ??

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    $\begingroup$ Note the 'only if'. All that you have to show is that a simple graph of strictly less than $5$ vertices cannot have $2n$ vertices. $\endgroup$ – Discrete lizard May 6 '18 at 16:47
  • $\begingroup$ Will do , thanks for your time, and you mean 2n edges right ? @Discretelizard $\endgroup$ – NANA May 6 '18 at 16:53
  • $\begingroup$ Indeed, I mean $2n$ edges, of course. $\endgroup$ – Discrete lizard May 6 '18 at 17:25
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When $n \leq 4$, the possible number of edges is less than twice the number of vertices. For example, when $n = 4$, there are 4 vertices but only 6 edges. In contrast, the complete graph on 5 vertices contains 10 edges.

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