1. **Problem Statement:** Show that a complete graph $K_n$ has a Hamiltonian cycle whenever $n \geq 3$. 2. **Definitions:** - A complete graph $K_n$ is a graph where every pair of distinct vertices is connected by a unique edge. - A Hamiltonian cycle is a cycle that visits every vertex exactly once and returns to the starting vertex. 3. **Key Idea:** In $K_n$, since every vertex is connected to every other vertex, it is always possible to form a cycle that includes all vertices. 4. **Proof:** - Since $K_n$ is complete, each vertex has degree $n-1$. - For $n \geq 3$, start at any vertex. - Because every vertex is connected to every other, you can travel from the starting vertex to any other vertex. - Visit each vertex exactly once by moving along edges connecting consecutive vertices. - Finally, return to the starting vertex to complete the cycle. 5. **Conclusion:** Thus, $K_n$ contains a Hamiltonian cycle for all $n \geq 3$. This is a fundamental property of complete graphs in graph theory.