Commun. Math. Res., 32 (2016), pp. 359-374.

Published online: 2021-05

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Let $G$ be a simple graph. A total coloring $f$ of $G$ is called an 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 an E-total coloring $f$ of a graph $G$ and any vertex $x$ of $G$, let $C(x)$ denote the set of colors of vertex $x$ and of the edges incident with , we call $C(x)$ the color set of $x$. If $C(u)≠C(v)$ 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 coloring of $G$ is denoted by $χ^e_{vt}(G)$ and is called the VDET chromatic number of $G$. The VDET coloring of complete bipartite graph $K_{7,n} (7 ≤ n ≤ 95)$ is discussed in this paper and the VDET chromatic number of $K_{7,n} (7 ≤ n ≤ 95)$ has been obtained.

}, issn = {2707-8523}, doi = {https://doi.org/10.13447/j.1674-5647.2016.04.08}, url = {http://global-sci.org/intro/article_detail/cmr/18908.html} }Let $G$ be a simple graph. A total coloring $f$ of $G$ is called an 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 an E-total coloring $f$ of a graph $G$ and any vertex $x$ of $G$, let $C(x)$ denote the set of colors of vertex $x$ and of the edges incident with , we call $C(x)$ the color set of $x$. If $C(u)≠C(v)$ 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 coloring of $G$ is denoted by $χ^e_{vt}(G)$ and is called the VDET chromatic number of $G$. The VDET coloring of complete bipartite graph $K_{7,n} (7 ≤ n ≤ 95)$ is discussed in this paper and the VDET chromatic number of $K_{7,n} (7 ≤ n ≤ 95)$ has been obtained.

*Communications in Mathematical Research*.

*32*(4). 359-374. doi:10.13447/j.1674-5647.2016.04.08