# Is $K_{3,3}$ with vertices on a circle locally rigid in the plane?

Consider the complete bipartite graph $$K_{3,3}$$ in plane such that all its vertices lie on a circle. Is this framework locally rigid in plane (which I believe is the case) and if so, how to prove this?

I know that the above framework is not infinitesimally rigid. Moreover, it is known (due to Bolker and Roth, 1980) that the above framework becomes infinitesimally rigid if the vertices do not lie on a conic.

So if the answer to the question is yes, then this would be an example of a framework in general position that is locally rigid but not infinitesimally rigid. This specific question was also asked earlier in MathOverflow (https://mathoverflow.net/questions/453909/frameworks-in-general-position-that-are-locally-rigid-but-not-infinitesimally-ri ) ,but didn't receive any answer.

• There are multiple ways to arrange $K_{3,3}$ so that all its vertices lie on a circle. Do you mean to put the vertices in a regular hexagon and connect the adjacent and the opposite vertices by edges? Sep 24, 2023 at 15:59
• @MishaLavrov The edge lengths should remain constant. Think of edges as bars and vertices as hinges. I am asking whether this bar framework can be flexed to a new non-congruent frmework having same edge lengths. Sep 24, 2023 at 16:16