Unsolved problems related to chromatic polynomials

For any simple graph G and any positive integer lambda, let

P(G,lambda) denote the number of mappings f from V(G) to

{1,2,..,lambda} such that f(u) not= f(v) for every two adjacent

vertices u and v in G. It can be shown that

P(G,lambda) = \sum_{A \subseteq E} (-1)^{|A|} lambda^{c(A)}

where E is the edge set of G and c(A) is the number of components

of the spanning subgraph of G with edge set A. Hence P(G,lambda)

is really a polynomial of lambda. Many results on the chromatic

polynomial of a graph have been discovered since it was introduced

by Birkhoff in 1912. However, there are still many unsolved

problems and this talk will introduce the progress of some

problems and also some new problems proposed recently.