Graphs with maximum degree D at least 17 and maximum average degree less than 3 are list 2-distance (D+2)-colorable

For graphs of bounded maximum average degree, we consider the problem of 2-distance coloring. This is the problem of coloring the vertices while ensuring that two vertices that are adjacent or have a common neighbor receive different colors. It is already known that planar graphs of girth at least 6 and of maximum degree∆ are list 2-distance (∆ + 2)-colorable when ∆ ≥ 24 (Borodin and Ivanova (2009)) and 2-distance (∆ + 2)-colorable when ∆ ≥ 18 (Borodin and Ivanova (2009)). We prove here that ∆ ≥ 17 suffices in both cases. More generally, we show that graphs with maximum average degree less than 3 and ∆ ≥ 17 are list 2-distance (∆ + 2)-colorable. The proof can be transposed to list injective (∆ + 1)-coloring.