Let $G=(V,E)$ be a finite simple graph such that between any two vertices there is a path of length at most $2$, and suppose that every vertex has at least $2$ neighbors.


Does $G$ have a Hamiltonian path (by which I mean a path visiting all vertices exactly once)?



Hamiltonian path in graphs of diameter and minimal degree $2$
**No.** Glue a bunch of triangles together at the same vertex. For a $2$-connected example, take the previous example and add a universal vertex. 


For an example that is $n$-connected take $K\_{n, n+2}$. 


