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

A proper [ k]-edge coloring of a graph G is a proper edge coloring of G using colors from $$[k]=\{1,2,\ldots ,k\}$$ . A neighbor sum distinguishing [ k]-edge coloring of G is a proper [ k]-edge coloring of G such that for each edge $$uv\in E(G)$$ , 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 $$G\ne C_5$$ , then nsdi $$(G)\le \varDelta (G)+2$$ . In this paper, we prove that this conjecture holds for $$K_4$$ -minor free graphs, moreover if $$\varDelta (G)\ge 5$$ , we show that nsdi $$(G)\le \varDelta (G)+1$$ . The bound $$\varDelta (G)+1$$ is sharp. [ABSTRACT FROM AUTHOR]