2-Distance vertex-distinguishing edge colorings of graphs
A 2-distance vertex-distinguishing edge coloring of a graph G is a proper edge coloring of G such that no two vertices at distance 2 get the same set of colors. The 2-distance vertex-distinguishing index χ′_d2(G) of a graph G is the minimum number of colors needed for a 2-distance vertex-distinguishing edge coloring of G. In this paper, we find the precise value of the 2-distance vertex-distinguishing indices for some graphs such as paths, cycles, trees, complete bipartite graphs and unicycle graphs. We also show that χ′_d2(G) ≤ Δ + 2 for a Halin graph with maximum degree Δ. An open problem is proposed.
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