Some problems on colouring and list colouring of planar graphs

A colouring of a graph $G$ is a mapping $\phi$ that assigns to each vertex $v$ a colour $\phi(v) \in \mathbb{N}$ such that adjacent vertices receive distinct colours. A list assignment for a graph $G$ is a mapping $L$ that assigns to each vertex $v$ a set $L(v)$ of permissible colours. An $L$-colouring of $G$ is a colouring $\phi$ with $\phi(v) \in L(v)$ for each vertex $v$. A planar graph is a graph $G$ that can be embedded on the plane, i.e., drawn in a piece of paper with with edge crossing. Colouring and list colouring of planar graphs is a subject studied extensively in the literature. Yet many challenging problems remains open. In this talk, I will survey some of these problems and current state of research on these problems.