How to personalize the vertices of a graph?

If f is a proper coloring of edges in a graph G = ( V , E ) , then for each vertex v ∈ V it defines the palette of colors of v , i.e., the set of colors of edges incident with v . In 1997, Burris and Schelp stated the following problem: how many colors do we have to use if we want to distinguish all vertices by their palettes. In general, we may need much more colors than χ ′ ( G ) . s paper we show that if we distinguish the vertices by color walks emanating from them, not just by their palettes, then the number of colors we need is very close to the chromatic index. Actually, not greater than Δ ( G ) + 1 .