Remarks on Vertex-Distinguishing IE-Total Coloring of Complete Bipartite Graphs $K_{4, n}$ and $K_{n, n}$

Abstract:

      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 $C(u)\neq C(v)$ 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 $\chi_{vt}^{ie}(G)$, and it is called the VDIET chromatic number of $G$. We will give VDIET chromatic numbers for complete bipartite graph $K_{4, n}$ $(n\ge 4)$, $K_{n, n}$ $(5\le n\le 21)$ in this article.