A list-assignment $L$ to the vertices of $G$ is the assignment of a list set $L(v)$ of colours to every vertex $v$ of $G$; and a $k$-list-assignment is a list-assignment such that $|L(v)|\geq k$, for every vertex $v$. If $L$ is a list-assignment to $G$, then an $L$-colouring of $G$ is a colouring (not necessarily proper) in which each vertex receives a colour from its own list;
The graph $G$ is $k$-list-colourable, or $k$-choosable, if it is properly $L$-colourable for every $k$-list-assignment $L$ to $G$. The chromatic number $\chi(G)$ of $G$ is the smallest number $k$ such that $G$ is $k$-colourable. The list chromatic number $\chi_l(G)$, is the smallest number $k$ such that $G$ is $k$-list-colourable or $k$-choosable.
It is evident that $\chi_l(G)\geq \chi(G)$, since if $k < \chi(G)$ then $G$ is not $L$-colourable when every vertex $v$ of $G$ is given the same list $L(v)$ of $k$ colours.
Consider the graph $S_n$ which has as vertex set the $n^2$ cells of our $n\times n$ array with two cells adjacent if and only if they are in the same row or column. The graph $S_n$ Since any $n$ cells in a row are pairwise adjacent we need at least $n$ colors. Furthermore, any coloring with $n$ colors corresponds to a Latin square, with the cells occupied by the same number forming a color class. Since Latin squares, as we have seen, exist, we infer $\chi(S_n) = n$, and the Dinitz problem can be stated as $$\chi_l(S_n) = n? $$
Now we know that the equality holds for all $n$. By the solution of Dinitz's problem, we know that the list chromatic number of $C_3\Box C_3$ is 3, i.e. $\chi_l(C_3\Box C_3)=3$.
The method of attack for the Dinitz problem is : We have to find an orientation of the graph $S_n$ with outdegrees $d_+(v) ≤ n−1$ for all $v$ and which ensures the existence of a kernel for all induced subgraphs.
I want to know the $\chi_l(C_3\Box C_5)$ and $\chi_l(C_5\Box C_5)$. I conjecure both of them are 3.
In the above oriented graph $C_3\Box C_5$, for all vetex $V$, it is easy to see that
$$ deg_-(v)=deg_+(v)=2.$$ I verified some of induced subgraph which indeed have a kernel. But I have no idea how to prove that all induced subgraph have a kernel. If someone can give any suggestions or comments, I will appreciate it. Thanks.