Abstract: Let f be a proper total k-coloring of a simple graph G. For any vertex x∈V(G), let Cf(x)denote the set of colors assigned to vertex x and the edges incident with x. If Cf(u)=Cf(v)for all distinct vertices u and v of V(G), then f is called a vertex-distinguishing total k-coloring of G. The minimum number k for which there exists a vertex-distinguishing total k-coloring of G is called the vertex-distinguishing total chromatic number of G and denoted by χ_vt(G). The vertex-disjoint union of two cycles of length n is denoted by 2Cn.We will obtain χ_vt(2Cn) in this paper. 

Key words: graphs, total coloring, vertex-distinguishing total coloring, vertex-distinguishing total chromatic number, cycle