Two-Distance Vertex-Distinguishing Index of Sparse Subcubic Graphs

Abstract

The 2-distance vertex-distinguishing index $$\chi '_\mathrm{d2}(G)$$ of a graph G is the minimum number of colors required for a proper edge coloring of G such that any pair of vertices at distance two have distinct sets of colors. It was conjectured that every subcubic graph G has $$\chi '_{\mathrm{d2}}(G)\le 5$$. In this paper, we confirm this conjecture for subcubic graphs with maximum average degree less than $$\frac{8}{3}$$.