how to show that list chromatic number $K_{n,n^n}$ is equal to $n+1$?
because $\chi_{L}(K_{n,m})\geq min(m,n)+1$ it is clear that it is at least $n+1$,how I should show that it is exactly $n+1$?
any hint ,ideas or reference will be great ,thanks a lot.
You wrote the easy inequality backwards. Trivially $\chi_L(K_{n,m})\le\min(m,n)+1$, so $\chi_L(K_{n,n^n})$ is at most $n+1$. (Just color the vertices on the small side first.) The trickier part is showing that $\chi_L(K_{n,n^n})\ge n+1$, i.e., that $K_{n,n^n}$ is not $n$-colorable. Hint: Assign disjoint lists of $n$ colors to the $n$ vertices on the left. For each vertex on the right, make a list consisting of one color from each list on the left; there are $n^n$ possible selections, assign a different one of these to each vertex on the right.