  The \emph{choice number} of a graph $G$, denoted $\ch(G)$, is the minimum
integer $k$ such that for any assignment of lists of size $k$ to the vertices
of $G$, there is a proper colouring of $G$ such that every vertex is mapped to
a colour in its list. For general graphs, the choice number is not bounded
above by a function of the chromatic number.
  In this thesis, we prove a conjecture of Ohba which asserts that
$\ch(G)=\chi(G)$ whenever $|V(G)|\leq 2\chi(G)+1$. We also prove a
strengthening of Ohba's Conjecture which is best possible for graphs on at most
$3\chi(G)$ vertices, and pose several conjectures related to our work.

Combinatorics;Discrete Mathematics