Small graph with gap between chromatic and vector chromatic number?
I make my comment an answer.
If we don't require that $\chi_v(G)$ is an integer then the smallest example is $G=C_5$ (a cycle on 5 vertices):
$$\chi_v(C_5) = \sqrt{5} &lt; 3 = \chi(C_5) . \qquad\text{[Lovász]}$$

It's not hard to transform this example to an example where $\chi_v(G)$ is integer. Let $G_1$ be a union of two 5-cycles $C_5^{(1)}$ and $C_5^{(2)}$ in which every vertex from $C_5^{(1)}$ is connected to every vertex in $C_5^{(2)}$. Let $G_2=K_5$. Finally, let $G$ be the union of $G_1$ and $G_2$. Then
\begin{align*}
\chi(G) &amp;= \max(\chi(G_1), \chi(G_2)) = \chi(G_1) = 6.\\
\chi_v(G) &amp;= \max(\chi_v(G_1), \chi_v(G_2)) = \max(2\cdot \sqrt{5}, 5) = 5.
\end{align*}