List coloring the square of sparse graphs with large degree

We consider the problem of coloring the squares of graphs of bounded maximum average degree, that is, the problem of coloring the vertices while ensuring that two vertices that are adjacent or have a common neighbour receive different colors. Borodin et al. proved in 2004 and 2008 that the squares of planar graphs of girth at least seven and sufficiently large maximum degree ∆ are list (∆ + 1)-colorable, while the squares of some planar graphs of girth six and arbitrarily large maximum degree are not. By Euler’s Formula, planar graphs of girth at least 6 are of maximum average degree less than 3, and planar graphs of girth at least 7 are of maximum average degree less than 14 5 < 3. We strengthen their result and prove that there exists a function f such that the square of any graphwithmaximum average degreem < 3 andmaximumdegree∆ ≥ f(m) is list (∆+1)colorable. This bound of 3 is optimal in the sense that the above-mentioned planar graphs with girth 6 havemaximum average degree less than 3 and arbitrarily largemaximum degree, while their square cannot be (∆ + 1)-colored. The same holds for list injective ∆-coloring.