# Show that if $|G| \geq 5$ and for each pair of vertices $u,v$ there is an $u−v$ Hamilton path then $\kappa(G) \geq 3$

Show that if $$G$$ is a graph with $$|G| \geq 5$$ such that for each pair of vertices $$u,v$$ there is an $$u−v$$ Hamilton path in $$G$$, then $$\kappa(G) \geq 3$$

I do not really have any idea how to start this task. Advices would be appreciated. Thanks.

Hmm, suppose there are $$u,v$$ such that $$G\setminus \{u,v\}$$ is disconnected. But there is a path $$u-v$$ through all the vertices of $$G$$ on it. So if we remove $$u,v$$ all the vertices are on the remainder of this path and there for connected. Have I missed something?
• Do we have to require $u$ and $v$ to be adjacent? – ensbana Mar 4 at 18:44
• No, why? $u$ and $v$ are arbitrary @ensbana – Maria Mazur Mar 4 at 18:45