Neighbor Sum Distinguishing Index of $$K_4$$K4-Minor Free Graphs

A proper [k]-edge coloring of a graph G is a proper edge coloring of G using colors from $$[k]=\{1,2,\ldots ,k\}$$[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 $$uv\in E(G)$$uv�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 with at least three vertices and $$G\ne C_5$$G�C5, then nsdi$$(G)\le \varDelta (G)+2$$(G)≤Δ(G)+2. In this paper, we prove that this conjecture holds for $$K_4$$K4-minor free graphs, moreover if $$\varDelta (G)\ge 5$$Δ(G)�5, we show that nsdi$$(G)\le \varDelta (G)+1$$(G)≤Δ(G)+1. The bound $$\varDelta (G)+1$$Δ(G)+1 is sharp.