Planar Graphs of Girth at least Five are Square $(\Delta + 2)$-Choosable

TL;DR

This paper proves that large girth planar graphs are square $( ext{max degree}+2)$-choosable and paintable, confirming a conjecture and extending coloring results to list and paint colorings.

Contribution

It establishes the square $( ext{max degree}+2)$-choosability and paintability of planar graphs with girth at least five, confirming a longstanding conjecture.

Findings

Planar graphs of girth at least five are square $( ext{max degree}+2)$-choosable.

Such graphs are also square $( ext{max degree}+2)$-paintable.

The result holds for sufficiently large maximum degree.

Abstract

We prove a conjecture of Dvo\v{r}\'ak, Kr\'al, Nejedl\'y, and \v{S}krekovski that planar graphs of girth at least five are square $(\Delta+2)$-colorable for large enough $\Delta$. In fact, we prove the stronger statement that such graphs are square $(\Delta+2)$-choosable and even square $(\Delta+2)$-paintable.