Perfect matching in a 2-regular graph

Is it true, that in every $2$-regular graph with $14$ vertices there is a perfect matching ? If you think it's true - prove it, otherwise show counter-example

This is my exercise. I think that it's true because it doesn't matter how do you draw this graph, it always will be a cycle, and because the cycle has even number of vertices there always will be a perfect matching.

• You might have left out an assumption of connectedness. – hardmath Oct 9 '15 at 22:29

HINT: Try making it a pair of $7$-cycles.

• oh my god. I am so embarrassed. Thank you. – Filip Kowalski Oct 9 '15 at 21:19
• @You’re welcome. (I think that we’ve all had that feeling a time or two.) – Brian M. Scott Oct 9 '15 at 21:22