The problem is the following.
Let $ G=(V,E)$ be a cycle of length $4n$ and let $$ V=V_1 \cup V_2\cup \cdots \cup V_n $$ be a partition of its $4n$ vertices into $n$ pairwise disjoint subsets, each of cardinality $4$. Is it true that there must be an independent set of $G$ containing precisely one vertex from each $V_i$ ? (Prove or supply a counter example.)
Actually, I could find a solution to this problem in MathStack but, I couldn't understand of that solution. You can see here. prove or supply counter example about graph
what I was not sure about is the following part. "there are only two vertices adjacent to either of the two vertices, implying that" at the last part of the proof. Why only vertices adjacent to 'two vertices' are concerned when we figure out the dependency? Could you elaborate on this?
P.S. I wanted to add comment to that post but I couldn't due to my lack of reputation. Sorry about that.