Vertex-Distinguishing E-Total Coloring of the Graphs $mC_{3}$ and $mC_{4}$

Abstract:

      Let $G$ be a simple graph. A total coloring $f$ of $G$ is called E-total-coloring if no two adjacent vertices of $G$ receive the same color and no edge of $G$ receives the same color as one of its endpoints. For E-total-coloring $f$ of a graph $G$ and any vertex $u$ of $G$, let $C_f(u)$ or $C(u)$ denote the set of colors of vertex $u$ and the edges incident to $u$. We call $C(u)$ the color set of $u$. If $C(u)\neq C(v)$ for any two different vertices $u$ and $v$ of $V(G)$, then we say that $f$ is a vertex-distinguishing E-total-coloring of $G$, or a $VDET$ coloring of $G$ for short. The minimum number of colors required for a $VDET$ colorings of $G$ is denoted by $\chi_{vt}^e(G)$, and it is called the VDET chromatic number of $G$. In this article, we will discuss vertex-distinguishing E-total colorings of the graphs $mC_{3}$ and $mC_{4}$.