Neighbor sum distinguishing index of planar graphs

A proper [ k ] -edge coloring of a graph G is a proper edge coloring of G using colors from [ k ] = { 1 , 2 , … , k } . A neighbor sum distinguishing [ k ] -edge coloring of G is a proper [ k ] -edge coloring of G such that for each edge u v ∈ E ( G ) , the sum of colors taken on the edges incident to u is different from the sum of colors taken on the edges incident to v . By nsdi ( G ) , we denote the smallest value k in such a coloring of G . It was conjectured by Flandrin et al. that if G is a connected graph without isolated edges and G ≠ C 5 , then nsdi ( G ) ≤ Δ ( G ) + 2 . In this paper, we show that if G is a planar graph without isolated edges, then nsdi ( G ) ≤ max { Δ ( G ) + 10 , 25 } , which improves the previous bound ( max { 2 Δ ( G ) + 1 , 25 } ) due to Dong and Wang.