Vertex-Distinguishing IE-Total Colorings of Cycles and Wheels (Communications in Mathematical Research, 2021)
Let $G$ be a simple graph. An IE-total coloring $f$ of $G$ refers to a coloringof 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 anIE-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 fora VDIET coloring of $G$ is denoted by $χ^{ie}_{vt}(G)$, and is called the VDIET chromaticnumber of $G$. We get the VDIET chromatic numbers of cycles and wheels, andpropose related conjectures in this paper.
Keywords: graph; IE-total coloring; vertex-distinguishing IE-total coloring; vertex-distinguishing IE-total chromatic number.