Local Neighbor-Distinguishing Index of Graphs

Suppose that G is a graph and ϕ is a proper edge-coloring of G . For a vertex v ∈ V ( G ) , let C ϕ ( v ) denote the set of colors assigned to the edges incident with v . The graph G is local neighbor-distinguishing with respect to the coloring ϕ if for any two adjacent vertices x and y of degree at least two, it holds that C ϕ ( x ) ⊈ C ϕ ( y ) and C ϕ ( y ) ⊈ C ϕ ( x ) . The local neighbor-distinguishing index, denoted χ lnd ′ ( G ) , of G is defined as the minimum number of colors in a local neighbor-distinguishing edge-coloring of G . For n ≥ 2 , let H n denote the graph obtained from the bipartite graph K 2 , n by inserting a 2-vertex into one edge. In this paper, we show the following results: (1) For any graph G , χ lnd ′ ( G ) ≤ 3 Δ - 1 ; (2) suppose that G is a planar graph. Then χ lnd ′ ( G ) ≤ ⌈ 2.8 Δ ⌉ + 4 ; and moreover χ lnd ′ ( G ) ≤ 2 Δ + 10 if G contains no 4-cycles; χ lnd ′ ( G ) ≤ Δ + 23 if G is 3-connected; and χ lnd ′ ( G ) ≤ Δ + 6 if G is Hamiltonian.