List $4$-colouring of planar graphs

Abstract

This paper proves the following result: If $G$ is a planar graph and $L$ is a $4$-list assignment of $G$ such that $|L(x) \cap L(y)| \le 2$ for every edge $xy$, then $G$ is $L$-colourable. This answers a question asked by Kratochv\'{i}l, Tuza and Voigt in [Journal of Graph Theory, 27(1):43--49, 1998].