# Hamiltonian path in graphs of diameter and minimal degree $2$

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)?

• The suspension of a discrete space which has at least $\ 4\$ different points (for a total of at least $\ 6\$ vertices for the whole graph) doesn't admit any Hamiltonian path. – Włodzimierz Holsztyński Mar 23 '17 at 13:16

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 $K_{n, n+2}$.