# Simple and 2-regular graph with exactly n vertices

How do i approach this problem:

For which natural numbers $n$ there exists a simple, 2-regular graph with exactly $n$ vertices?

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## 1 Answer

Hint: I believe that a simple, 2-regular graph with exactly n vertices is called an n-cycle.

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Thank you!. How about simple, 3-regular graph with exactly n vertices? –  jenn3 Jan 31 '13 at 5:21
@jenn3 That's a separate question, but not hard either. –  Calvin Lin Jan 31 '13 at 5:22
would this work for every n? how do i show that –  jenn3 Jan 31 '13 at 5:35
The logic will work for every single $n$. For example, it suffices to consider: is there a 0-cylce? How about a 1-cycle? How about a 2-cycle? –  Calvin Lin Jan 31 '13 at 5:40
Thanks again. For simple, 3-regular graph is basically called a cubic graph right? which exist for n >= 4 and even. –  jenn3 Jan 31 '13 at 6:06