consider the bipartite graph shown below,how should I show that list chromatic number for this graph is 3?
because it is bipartite the chromatic number is 2,and because $\chi _{L}(G) \geq \chi(G)$ for all graphs,so its list chromatic number is at least 2,and as you see the list that I assign to each vertex,it can't be colored such this way so it couldn't be 2,so it is at least 3,I think checking that all list of length 3 is not good Idea,what should I do? thanks.
First color the two vertices of degree $3$. Then color the remaining vertices; there will be no problem because they all have degree $2$.
More generally, if the vertices of a graph can be sequenced so that each vertex is adjacent to at most two of the preceding vertices, then the graph is list $3$-colorable. (Graphs with such a sequencing are called $2$-degenerate.) If the graph has only two vertices of degree $\ge3$ it's easy, just put those vertices first.