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The task in the image is to prove the closure lemma, which is typically associated with Hamiltonian cycles in graph theory. The lemma states:

Let \( G \) be a graph, and let \( ab \notin E(G) \) (i.e., the edge between vertices \( a \) and \( b \) is not in the graph). Suppose that the degree of vertex \( a \) plus the degree of vertex \( b \) is at least the number of vertices in the graph, i.e.,

\[
d_G(a) + d_G(b) \geq v(G)
\]

Then, the graph \( G \) contains a Hamiltonian cycle if and only if the graph \( G + ab \) (which is the graph \( G \) with the edge \( ab \) added) contains a Hamiltonian cycle.

### Proof Outline:
1. **Graph Concepts**: We work with a graph \( G \) that might not contain a Hamiltonian cycle. The idea is to investigate what happens when we add an edge \( ab \).
2. **Condition on Degrees**: The condition \( d_G(a) + d_G(b) \geq v(G) \) suggests that the combined degree of vertices \( a \) and \( b \) is sufficient to potentially "force" a Hamiltonian cycle if this edge \( ab \) were added.
3. **Closure**: If adding \( ab \) forms a Hamiltonian cycle, then we know \( G \) also had the necessary structure to support one, even before adding \( ab \). Conversely, if \( G \) already contains a Hamiltonian cycle, adding this edge does not change that fact.
4. **Conclusion**: Thus, the graph \( G \) contains a Hamiltonian cycle if and only if the graph \( G + ab \) does.

### Do you need further steps in the proof or clarification? 

#### Additional related questions:
1. What is a Hamiltonian cycle, and why is it significant in graph theory?
2. How does the degree condition \( d_G(a) + d_G(b) \geq v(G) \) influence the existence of a Hamiltonian cycle?
3. What is the role of the closure of a graph in this lemma?
4. How do you formally prove the necessity and sufficiency parts of this lemma?
5. Can you give an example of a graph where this lemma applies?

#### Tip:
Understanding the definition and properties of a Hamiltonian cycle is key to mastering this type of problem. A Hamiltonian cycle is a cycle that visits every vertex exactly once and returns to the starting vertex.

Докажите лемму о замыкании: пусть G — граф, ab ∉ E(G), d_G(a) + d_G(b) ≥ v(G). Тогда граф G содержит гамильтонов цикл, если и только если граф G + ab его содержит.

Learn how to prove the closure lemma in graph theory, which establishes that a graph G contains a Hamiltonian cycle if and only if the graph G + ab contains one, under specific degree conditions. Suitable for advanced university-level study.

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