On the Neighbor-Distinguishing Indices of Planar Graphs

Abstract

Let G be a simple graph with no isolated edges. The neighbor-distinguishing edge coloring of G is a proper edge coloring of G such that any pair of adjacent vertices have different sets consisting of colors assigned on their incident edges. The neighbor-distinguishing index of G, denoted by χa′(G), is the minimum number of colors in such an edge coloring of G. In this paper, we show that if G is a connected planar graph with maximum degree Δ ≥ 14 , then Δ≤χa′(G)≤Δ+1, and χa′(G)=Δ+1 if and only if G contains a pair of adjacent vertices of maximum degree. This improves a result in [W. Wang, D. Huang, A characterization on the adjacent vertex distinguishing index of planar graphs with large maximum degree, SIAM J. Discrete Math. 29(2015), 2412–2431], which says that every connected planar graph G with Δ ≥ 16 has Δ≤χa′(G)≤Δ+1, and χa′(G)=Δ+1 if and only if G contains a pair of adjacent vertices of maximum degree.