Let $G$ be a simple graph. An IE-total coloring $f$ of $G$ refers to a coloring
of the vertices and edges of $G$ so that no two adjacent vertices receive the same color.
Let $C(u)$ be the set of colors of vertex $u$ and edges incident to $u$ under $f$. For an
IE-total coloring $f$ of $G$ using $k$ colors, if $C(u)≠C(v)$ for any two different vertices $u$ and $v$ of $V (G)$, then $f$ is called a $k$-vertex-distinguishing IE-total-coloring of $G$,
or a $k$-VDIET coloring of $G$ for short. The minimum number of colors required for
a VDIET coloring of $G$ is denoted by $χ^{ie}_{vt}(G)$, and is called the VDIET chromatic
number of $G$. We get the VDIET chromatic numbers of cycles and wheels, and
propose related conjectures in this paper.
