List $4$-colouring of planar graphs

TL;DR

This paper proves a new list-coloring theorem for planar graphs, showing they are colorable under specific list intersection constraints, thus answering a longstanding open question in graph theory.

Contribution

It establishes a novel 4-list coloring result for planar graphs with limited list intersection, advancing understanding of graph coloring constraints.

Findings

Proves that planar graphs are L-colorable under the given conditions.

Answers an open question posed in 1998.

Provides a new bound on list intersections for colorability.

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].