Multiple list colouring of planar graphs

TL;DR

This paper demonstrates the limitations of multiple list coloring in planar graphs by constructing specific counterexamples and proposes conjectures to guide future research in this area.

Contribution

It introduces new counterexamples showing certain multiple list colorings are impossible in planar graphs and formulates conjectures for further investigation.

Findings

Existence of planar graphs not $(4m+loor{rac{2m-1}{9}},m)$-choosable for each positive integer m

Counterexamples challenge previous assumptions about multiple list coloring

Proposed conjectures to explore the boundaries of multiple list coloring in planar graphs

Abstract

This paper proves that for each positive integer $m$, there is a planar graph $G$ which is not $(4m+\lfloor \frac{2m-1}{9}\rfloor,m)$-choosable. Then we pose some conjectures concerning multiple list colouring of planar graphs.