Neighbor-sum-distinguishing edge choosability of subcubic graphs

A graph G is said to be neighbor-sum-distinguishing edge k-choose if, for every list L of colors such that L(e) is a set of k positive real numbers for every edge e, there exists a proper edge coloring which assigns to each edge a color from its list so that for each pair of adjacent vertices u and v the sum of colors taken on the edges incident to u is different from the sum of colors taken on the edges incident to v. Let $$\mathrm{ch}^{\prime }_{\sum ^p}(G)$$ ch ∑ p ′ ( G ) denote the smallest integer k such that G is neighbor-sum-distinguishing edge k-choose. In this paper, we prove that if G is a subcubic graph with the maximum average degree mad(G), then (1) $$\mathrm{ch}^{\prime }_{\sum ^p}(G)\le 7$$ ch ∑ p ′ ( G ) ≤ 7 ; (2) $$\mathrm{ch}^{\prime }_{\sum ^p}(G)\le 6$$ ch ∑ p ′ ( G ) ≤ 6 if $$\hbox {mad}(G)