Abstract: Let G be a simple graph. A general edge-coloring of a graph G is an assignment of a number of colors to the edges. It is not necessary to assign two distinct colors to two adjacent edges. A general edge-coloring f of a graph G is called vertex distinguished by multisets, if, for any two distinct vertices u,v of a graph G, the multiset of the colors used to color the edges incident with u is different from the multiset of the colors used to color the edges incident with v. The minimum number of colors required for a general edge-coloring of G which is vertex distinguishing by multisets, denoted by c(G), is called the vertex distinguishing general edge chromatic number of G by multisets. Suppose mP_n denotes the vertex-disjoint union of m paths of length n. The vertex distinguishing general edge-coloring(by multisets) of mP_n will be discussed.

Key words: general edge-coloring, vertex distinguished by multisets, path, vertex-disjoint union