# (4, 2)-Choosability of Planar Graphs with Forbidden Structures

## Abstract

All planar graphs are 4-colorable and 5-choosable, while some planar graphs are not 4-choosable. Determining which properties guarantee that a planar graph can be colored using lists of size four has received significant attention. In terms of constraining the structure of the graph, for any \(\ell \in \{3,4,5,6,7\}\), a planar graph is 4-choosable if it is \(\ell \)-cycle-free. In terms of constraining the list assignment, one refinement of *k*-choosability is *choosability with separation*. A graph is *(k, s)-choosable* if the graph is colorable from lists of size *k* where adjacent vertices have at most *s* common colors in their lists. Every planar graph is (4, 1)-choosable, but there exist planar graphs that are not (4, 3)-choosable. It is an open question whether planar graphs are always (4, 2)-choosable. A *chorded* \(\ell \)-*cycle* is an \(\ell \)-cycle with one additional edge. We demonstrate for each \(\ell \in \{5,6,7\}\) that a planar graph is (4, 2)-choosable if it does not contain chorded \(\ell \)-cycles.

## Keywords

Graph coloring Planar graph Choosability with separation Discharging## Notes

### Acknowledgements

We are grateful to anonymous referee for spotting mistakes in the previous version of the manuscript. We thank Ryan R. Martin, Alex Nowak, Alex Schulte, and Shanise Walker for participation in the early stages of the Project.

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