1. Prove that when m\geq k^{k}, the list-chromatic number of K_{m,k} exceeds k.

2. Let f be a proper k-coloring of a k-chromatic graph G. Let T be a tree with vertex set \{w_1, w_2,..., w_k\}. Prove that there is an edge-preserving map \phi : V(T) \rightarrow V(G) such that f(\phi(w_i)) = i for all i.

3. Prove that K_{m,n} has an interval coloring using m + n - gcd(m,n) colors, and that fewer colors cannot suffice.

4. Prove that \chi_L (K_{2,...,2,3})= r, where there are r-1 2's in the subscript of K.

I have worked other problems, but I am completely stuck on these 4. Thanks.