Coloring the square of a sparse graph G with almost Δ(G) colors

For a graph G, let G2 be the graph with the same vertex set as G and xy∈E(G2) when x≠y and dG(x,y)≤2. Bonamy, Lévêque, and Pinlou conjectured that if mad(G)<4−2c+1 and Δ(G) is large, then χℓ(G2)≤Δ(G)+c. We prove that if c≥3, mad(G)<4−4c+1, and Δ(G) is large, then χℓ(G2)≤Δ(G)+c. Dvořák, Král', Nejedlý, and Å krekovski conjectured that χ(G2)≤Δ(G)+2 when Δ(G) is large and G is planar with girth at least 5; our result implies χ(G2)≤Δ(G)+6.