Let $G(V, E)$ be a simple, undirected graph. A vertex cover of $G$ is a subset $V^{\prime} \subseteq V$ such that for every $(u, v) \in E, u \in V^{\prime \prime}$ or $v \in V^{\prime}$. Let the size of the smallest vertex cover in $G$ be $k$. Let $S$ be any vertex cover of size $k$.

For a vertex $v \in V$, which of the following constraints will always ensure that $v \in S$ ?

Let $G$ be an undirected graph, which is a path on 8 vertices. The number of matchings in $G$ is $\_\_\_\_$ (answer in integer)

The chromatic number of a graph is the minimum number of colours used in a proper colouring of the graph. The chromatic number of the following graph is ________

The chromatic number of a graph is the minimum number of colours used in a proper colouring of the graph. Let $G$ be any graph with $n$ vertices and chromatic number $k$. Which of the following statements is/are always TRUE?