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

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.
Comment: 22 pages, 7 figures; this version incorporates reviewer feedback: fixed and rewrote proof of Lemma 3.5, expanded Section 4; to appear in J. Combin. Theory Ser. B