On choosability with separation of planar graphs with lists of different sizes

TL;DR

This paper investigates the choosability with separation of planar graphs, specifically strengthening known results by allowing an independent set of vertices to have smaller lists, advancing understanding of list coloring constraints.

Contribution

It extends the known (4,1)-choosability of planar graphs by permitting an independent set to have lists of size 3, improving previous bounds.

Findings

Planar graphs are (4,1)-choosable.

Allowing an independent set to have lists of size 3 is sufficient.

Strengthens existing choosability results for planar graphs.

Abstract

A (k,d)-list assignment L of a graph G is a mapping that assigns to each vertex v a list L(v) of at least k colors and for any adjacent pair xy, the lists L(x) and L(y) share at most d colors. A graph G is (k,d)-choosable if there exists an L-coloring of G for every (k,d)-list assignment L. This concept is also known as choosability with separation. It is known that planar graphs are (4,1)-choosable but it is not known if planar graphs are (3,1)-choosable. We strengthen the result that planar graphs are (4,1)-choosable by allowing an independent set of vertices to have lists of size 3 instead of 4.