Strict Neighbor-Distinguishing Index of Subcubic Graphs

A proper edge coloring of a graph G is strict neighbor-distinguishing if for any two adjacent vertices u and v, the set of colors used on the edges incident to u and the set of colors used on the edges incident to v are not included with each other. The strict neighbor-distinguishing index of G is the minimum number $$\chi '_\mathrm{snd}(G)$$ of colors in a strict neighbor-distinguishing edge coloring of G. In this paper, we prove that every connected subcubic graph G with $$\delta (G)\ge 2$$ has $$\chi '_\mathrm{snd}(G)\le 7$$ , and moreover $$\chi '_\mathrm{snd}(G)=7$$ if and only if G is a graph obtained from the graph $$K_{2,3}$$ by inserting a 2-vertex into one edge.