Sufficient sparseness conditions for G² to be (Delta+1)-choosable, when Deltage5

We determine the list chromatic number of the square of a graph $\chil(G^2)$ in terms of its maximum degree $\Delta$ when its maximum average degree, denoted $\mad(G)$, is sufficiently small. For $\Delta\ge 6$, if $\mad(G)<2+\frac{4\Delta-8}{5\Delta+2}$, then $\chil(G^2)=\Delta+1$. In particular, if $G$ is planar with girth $g\ge 7+\frac{12}{\Delta-2}$, then $\chil(G^2)=\Delta+1$. Under the same conditions, $\chil^i(G)=\Delta$, where $\chil^i$ is the list injective chromatic number.