More specifically, does the stronger statement apply (Georgakopoulos 2009)?
"If $G=(V,E)$ is a 2-connected ﬁnite graph and $x \in V(G)$, then $G^2$ has a Hamilton cycle whose edges at $x$ lie in $E(G)$."
I am not sure I understand that statement fully. In the graph $G$ below the line $ag$ (or some other grey line) does not lie in $E(G)$, and I assume the existence of Hamilton cycle requires that line in G^2.
I either proved Georgakopoulos wrong (yeah right) or misunderstood something badly. I got wrong the statement, hamiltonicity, definition of $G^2$, or something else.
The graph in question
$G$ (black) and $G^2$ (black and grey):