Consider an undirected graph $G=(V,E)$. The maximum degree of any vertex is $10$. A set $E'\subseteq E$ is called "maximal independent" if no two edges in $E'$ share a vertex, but adding any more edge to $E'$ breaks this property. An "operation" is to select any maximal independent set of the set of remaining edges, and delete the edges in that set. We can keep performing the operations until there are no edges left.
What is the maximal number of operations we might be able to perform?
HINT: In what kind of graph is every independent set a singleton? In such graphs each step will remove only one vertex, so it would be reasonable to conjecture that they require the largest number of operations for a given maximal degree. You should at least be able to formulate this conjecture.
Proving that the conjecture is correct is harder, but it can be done by induction on $n$, the maximal degree of a vertex of $G$, if you notice that removing a maximal independent set always reduces the maximal degree. (Why?)