What is an example of a graph with chromatic number $\chi(G)=3$ and list-chromatic number $\chi_\ell(G)=4$?
My first thought was to consider complete tripartite graphs since these will have chromatic number $3$. I also know that $K_{3,3}$ has list chromatic number $3$, so tried (without success) to find a $3$-list-assignment for $K_{3,3,1}$ which was not colourable.
Let $G$ be a graph with chromatic number $2$ and list chromatic number $3,$ e.g. $G=K_{3,3},$ or just take $C_6$ and add an edge joining two diametrically opposite vertices.
Take the graph $3G$ (the union of three vertex-disjoint copies of $G$); add a new vertex $v$ and edges joining $v$ to every vertex of $3G.$ You can easily show that the resulting graph has chromatic number $3$ and list chromatic number $4.$