Undirected graph with 12 edges and 6 vertices [closed]

For school we have to make an assignment, and part of the assignment is this question:

Describe an unidrected graph that has 12 edges and at least 6 vertices. 6 of the vertices have to have degree exactly 3, all other vertices have to have degree less than 2. Use as few vertices as possible.

The best solution I came up with is the following one. Here the number in the circles is the degree of that vertex, now I was wondering if there is a better solution, if so, can somebody explain this to me?

I do not need a better answer, just a push in the right direction - if needed.

closed as off-topic by Juho, vonbrand, G. Bach, David Richerby, Artem KaznatcheevMar 27 '14 at 7:58

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• Your graph has only $11$ edges. It's pretty obvious where to put the last edge. Also, considering $\sum_{v \in V} \deg(v) = 2m$, you can't do better than your graph given the restrictions you have to observe. – G. Bach Mar 23 '14 at 16:16