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Well, here's what sage-5.0-beta1 has to say on the instance :-p

sage: H=Graph([(0, 1), (0, 3), (0, 4), (0, 5), (1, 2), (1, 3), (1, 4), (1, 6), (2, 5), (3, 4), (5, 6)])
sage: H.girth()
4
sage: H.is_triangle_free()
False

It should notice it is saying something stupid, only it does not. Sage is not a very bright student :-p