Summary: We give nontrivial bounds for the inductiveness or degeneracy of power graphs \(G^{k}\) of a planar graph \(G\). This implies bounds for the chromatic number as well, since the inductiveness naturally relates to a greedy algorithm for vertex-coloring the given graph. The inductiveness moreover yields bounds for the choosability of the graph. We show that the inductiveness of a square of a planar graph \(G\) is at most \(\lceil 9\Delta /5 \rceil\), for the maximum degree \(\Delta\) sufficiently large, and that it is sharp. In general, we show for a fixed integer \(k\geq 1\) the inductiveness, the chromatic number, and the choosability of \(G^{k}\) to be \(O(\Delta^{\lfloor k/2 \rfloor})\), which is tight.