Commun. Math. Res., 30 (2014), pp. 222-236.
Published online: 2021-05
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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.
}, issn = {2707-8523}, doi = {https://doi.org/10.13447/j.1674-5647.2014.03.04}, url = {http://global-sci.org/intro/article_detail/cmr/18965.html} }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.