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"Under what conditions does a rotationally symmetric (undirected) 2D graph have >=1 rotationally symmetric Eulerian circuit?

I'm sure it's possible to generalize rotational symmetry to graphs as a whole, but I'm not the mathematician to do it, so my focus is on graphs whose nodes are plotted at specific points in a 2D space.

Attached are two rotationally symmetric graphs; one definitely has a rotationally symmetric Eulerian circuit. The other I've looked & I can't find one."