Sufficient sparseness conditions for G2 to be (Δ+1)-choosable, when Δ≥5

We determine the list chromatic number of the square of a graph χl(G 2) in terms of its maximum degree ∆ when its maximum average degree, denoted mad(G), is sufficiently small. For ∆ ≥ 6, if mad(G) < 2 + 4∆−8 5∆+2 , then χl(G 2) = ∆ + 1. In particular, if G is planar with girth g ≥ 7 + 12 ∆−2 , then χl(G 2) = ∆ + 1. Under the same conditions, χil(G) = ∆, where χ i l is the list injective chromatic number.