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.