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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 – Yemon Choi Dec 25 '10 at 17:41
Or, if this is not homework/coursework, see – 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

You can find an example of a bridgeless graph with cut points, whose square is not hamiltonian in this paper of Fleischner and Kronk. (I know the paper is in German, but the figure of the graph is on the first page.) Fleischner also mentions this example in his paper "The Square of Every Two-Connected Graph Is Hamiltonian".

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