Chromatic number of a graph with $10$ vertices each of degree $8$?
Let $G = (V, E)$ be a simple graph like you described (which is unique up to isomorphism and known as the Turán graph $T(10, 5)$, by the way). Note that the maximal degree possible in a graph with $10$ vertices is $9$ and thus, for every vertex $v$ in $G$ there exists a unique vertex $w \ne v$ which is not connected to $v$ and the two vertices share a neighborhood, i.e. $N(v) = N(w)$. Therefore, $v$ and $w$ may be colored using the same color. This however implies that the chromatic number of $G$, denoted by $\chi(G)$, satisfies $\chi(G) \leq 5$ since we can take all such pairs $\{v, w\}$ as described above and assign every such pair a unique color (using 5 colors in total).

To see that $\chi(G)$ cannot be less than $5$ consider the (induced) subgraph $G'$ of $G$ that we get by removing one vertex of every such pair $\{v, w\}$ as described above (that means $vw \notin E$). We find that $G'$ consists of 5 vertices and is complete, therefore implying that $G$ contains a clique of size $5$ and giving us $\chi(G) \geq 5$.

Hence we have shown $\chi(G) = 5$.