List-coloring the square of a subcubic graph

The square G2 of a graph G is the graph with the same vertex set as G and with two vertices adjacent if their distance in G is at most 2. Thomassen showed that for a planar graph G with maximum degree ∆(G) = 3 we have χ(G2) ≤ 7. Kostochka and Woodall conjectured that for every graph, the list-chromatic number of G2 equals the chromatic number of G2, that is χl(G 2) = χ(G2) for all G. If true, this conjecture (together with Thomassen’s result) implies that every planar graph G with ∆(G) = 3 satisfies χl(G 2) ≤ 7. We prove that every graph (not necessarily planar) with ∆(G) = 3 other than the Petersen graph satisfies χl(G 2) ≤ 8 (and this is best possible). In addition, we show that if G is a planar graph with ∆(G) = 3 and girth g(G) ≥ 7, then χl(G 2) ≤ 7. Dvor̆ák, S̆krekovski, and Tancer showed that if G is a planar graph with ∆(G) = 3 and girth g(G) ≥ 10, then χl(G 2) ≤ 6. We improve the girth bound to show that if G is a planar graph with ∆(G) = 3 and g(G) ≥ 9, then χl(G 2) ≤ 6. All of our proofs can be easily translated into linear-time coloring algorithms.