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Volume 32, Issue 4
Vertex-Distinguishing E-Total Coloring of Complete Bipartite Graph $K_{7,n}$ when $7 ≤ n ≤ 95$

Xiang-En Chen

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

Published online: 2021-05

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  • Abstract

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.

  • Keywords

graph, complete bipartite graph, E-total coloring, vertex-distinguishing E-total coloring, vertex-distinguishing E-total chromatic number.

  • AMS Subject Headings

05C15

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CMR-32-359, author = {Xiang-En and Chen and and 18586 and and Xiang-En Chen}, title = {Vertex-Distinguishing E-Total Coloring of Complete Bipartite Graph $K_{7,n}$ when $7 ≤ n ≤ 95$}, journal = {Communications in Mathematical Research }, year = {2021}, volume = {32}, number = {4}, pages = {359--374}, abstract = {

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} }
TY - JOUR T1 - Vertex-Distinguishing E-Total Coloring of Complete Bipartite Graph $K_{7,n}$ when $7 ≤ n ≤ 95$ AU - Chen , Xiang-En JO - Communications in Mathematical Research VL - 4 SP - 359 EP - 374 PY - 2021 DA - 2021/05 SN - 32 DO - http://doi.org/10.13447/j.1674-5647.2016.04.08 UR - https://global-sci.org/intro/article_detail/cmr/18908.html KW - graph, complete bipartite graph, E-total coloring, vertex-distinguishing E-total coloring, vertex-distinguishing E-total chromatic number. AB -

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.

Xiang-En Chen. (2021). Vertex-Distinguishing E-Total Coloring of Complete Bipartite Graph $K_{7,n}$ when $7 ≤ n ≤ 95$. Communications in Mathematical Research . 32 (4). 359-374. doi:10.13447/j.1674-5647.2016.04.08
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