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It is easy to see that any (locally finite) graph is homotopy-equivalent to a trivalent graph. Moreover, this can be achieved by a local construction - take neighborhoods of vertices of degree $> 3$...

I have frequently come across the statement "Any two triangulations of a compact n-manifold are related by bistellar moves" attributed to Pachner via Lickorish's paper 'Simplicial moves on complexes ...