# Wanted: a graph $G$ without bridges, whose square is not hamiltonian

Construct an example of graph $G$ without bridges, such that its square $G^2$ is non hamiltonian. Note: Since Fleischner's Theorem (the square of each 2-connected graph is Hamiltonian) and bridges are forbidden, the required graph should have at least one cut-vertex.

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Please see mathoverflow.net/faq#whatnot –  Yemon Choi Dec 25 '10 at 17:41
Or, if this is not homework/coursework, see mathoverflow.net/howtoask –  Yemon Choi Dec 25 '10 at 17:42
No, the teacher said that this example exists, but he did not remember it. He also said, that as conclusion we obtain that the Fleischner's Theorem does not improve. –  Michael Dec 25 '10 at 17:44
I tried to find information on the Internet, but had no success –  Michael Dec 25 '10 at 17:51
In this context the square of a graph $G$ has the same vertices but has edges between vertices if their distance in $G$ is 1 or 2. –  Aaron Meyerowitz Dec 26 '10 at 6:30