Let's proceed step by step: --- **Step 1: Recall Dirac's Theorem** **Statement:** Dirac's Theorem says that if \( G \) is a simple graph with \( n \geq 3 \) vertices and **minimum degree** \(\delta(G) \geq \frac{n}{2}\), then \( G \) is **Hamiltonian** (contains a Hamiltonian cycle). **Explanation:** This theorem gives a sufficient condition for Hamiltonicity based on the minimum degree. If every vertex is "well-connected" (at least half the vertices), then the graph must contain a cycle that visits every vertex exactly once. --- **Step 2: Relate the Given Condition to Dirac's Theorem** Given: \( n \leq 2\delta(G) \) Rewrite: \[ n \leq 2\delta(G) \implies \frac{n}{2} \leq \delta(G) \] Thus, the minimum degree \(\delta(G)\) is at least \(\frac{n}{2}\), which is **exactly** the condition required by Dirac's Theorem. **Explanation:** The given inequality ensures that the minimum degree in the graph is enough to guarantee a Hamiltonian cycle by Dirac's Theorem. --- **Final Answer:** By Dirac's Theorem, since \( \delta(G) \geq \frac{n}{2} \), the graph \( G \) is **Hamiltonian**.

Let's proceed with a detailed two-step solution: --- **Step 1: Understand the given condition and recall Dirac's Theorem** **Explanation:** Dirac's Theorem states that if a simple graph \( G \) with \( n \geq 3 \) vertices has a minimum degree \(\delta(G) \geq \frac{n}{2}\), then \( G \) is Hamiltonian. Our goal is to see if the given condition \( n \leq 2 \delta(G) \) guarantees this condition. --- **Step 2: Show that \( n \leq 2 \delta(G) \) implies \(\delta(G) \geq \frac{n}{2}\), and then apply Dirac's Theorem** **Explanation:** Starting from the inequality: \[ n \leq 2 \delta(G) \implies \frac{n}{2} \leq \delta(G) \] Since the minimum degree \(\delta(G)\) is at least \(\frac{n}{2}\), the conditions of Dirac's Theorem are satisfied. **Conclusion:** Therefore, by Dirac's Theorem, \( G \) is Hamiltonian. --- **Final Answer:** Given \( n \leq 2 \delta(G) \), we have \(\delta(G) \geq \frac{n}{2}\). By Dirac's Theorem, \( G \) contains a Hamiltonian cycle.