Graphs with maximum average degree less than 114 are (1,3)-choosable

The well known 1-2-3-Conjecture asserts that every connected graph G with at least three vertices can be edge weighted with 1,2,3, so that for any two adjacent vertices u and v, the sum of the weights of the edges incident to u is distinct from the sum of the weights of the edges incident to v. In this paper, we consider the list version of this problem and prove that graphs with maximum average degree smaller than 114 are strongly (1,3)-choosable, which implies that the 1-2-3 conjecture is true for such graphs. This improves the results in Cranston et al. (2014)[7] and Przybyło et al. (2017).