An improved upper bound for the neighbor sum distinguishing index of graphs ?
A proper [k] [ k ] -edge coloring of a graph G G is a proper edge coloring of G G using colors from [k]={1,2,,k} [ k ] = { 1 , 2 ,  , k } . A neighbor sum distinguishing [k] [ k ] -edge coloring of G G is a proper [k] [ k ] -edge coloring of G G such that for each edge uv?E(G) u v ? E ( G ) , the sum of colors taken on the edges incident to u u is different from the sum of colors taken on the edges incident to v v . By ? ? ? ( G ) , we denote the smallest value k k in such a coloring of G G . It was conjectured by Flandrin etal. that if G G is a connected graph with at least three vertices and G?C5 G ? C 5 , then ? ? ? ( G ) ? ? ( G ) + 2 . They gave an upper bound : ? 7 ? ( G ) ? 4 2 ? when ?(G)?2 ? ( G ) ? 2 . In this paper, we show that ? ? ? ( G ) ? ? 10 ? ( G ) + 2 3 ? , which improves the previous bound when ?(G)?18 ? ( G ) ? 18 .
neighbor sum distinguishing neighbor distinguishing