Question 4: Let G be a simple, undirected, unweighted graph. We use V…

Let $G$ be a simple, undirected, unweighted graph.

We use $V(G)$ to denote the vertex set of $G$ and $E(G)$ to denote the edge set of $G$.

Recall that if we have an empty graph (i.e, a graph where $V(G) = E(G) = \emptyset$), then by convention it (vacuously) satisfies the properties suggested in the given subquestions

Consider the set system $(U, \mathcal{F})$ defined as follows.

• The universe $U$ is $V(G)$.
• A subset of vertices $S \subseteq V(G)$ belongs to $\mathcal{F}$ if the subgraph induced by $S$ has minimum degree three.

Which of the following properties is/are NOT satisfied by $\mathcal{F}$?