Locally planar graphs are 5-paintable

A graph GG is locally planar if it is embedded in a surface with large edge-width. Thomassen (1993) proved that every graph embedded in a fixed surface with sufficiently large edge-width is 5-colourable. DeVos et al. (2008) strengthened this result and proved that every graph embedded in a fixed surface with sufficiently large edge-width is 5-choosable. This paper further strengthens the result to on-line list colouring and proves that every graph embedded in a fixed surface with sufficiently large edge-width is 5-paintable.