VN-TST 1998 Problem 6

In a conference there are $n \geq 10$ people. It is known that:

I. Each person knows at least $\left[\frac{n+2}{3}\right]$ other people. II. For each pair of person $A$ and $B$ who don't know each other, there exist some people $A(1), A(2), \ldots, A(k)$ such that $A$ knows $A(1)$ , $A(i)$ knows $A(i+1)$ and $A(k)$ knows $B$ . III. There doesn't exist a Hamilton path.

Prove that: We can divide those people into 2 groups: $A$ group has a Hamilton cycle, and the other contains of people who don't know each other.