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 ________

A graph has a chromatic number of $2$ if and only if it is bipartite and contains at least one edge (i.e., it contains no odd-length cycles).

1. Let us partition the $8$ vertices into two disjoint sets $V_1$ and $V_2$ of $4$ vertices each.

2. By observing the graph structure, no two vertices in $V_1$ share an edge, and no two vertices in $V_2$ share an edge. All edges in the graph only run between vertices in $V_1$ and vertices in $V_2$.

3. Since the graph is bipartite and contains edges, it can be properly colored using only $2$ colors (assigning color 1 to all vertices in $V_1$ and color 2 to all vertices in $V_2$).

4. Since the graph contains edges, it cannot be colored with 1 color.

Thus, the chromatic number $\chi(G) = 2$.