ON LOCAL ANTIMAGIC CHROMATIC NUMBER OF GRAPHS | ||
| Journal of Algebraic Systems | ||
| مقاله 11، دوره 7، شماره 2، فروردین 2020، صفحه 245-256 اصل مقاله (107.64 K) | ||
| نوع مقاله: Original Manuscript | ||
| شناسه دیجیتال (DOI): 10.22044/jas.2019.7933.1391 | ||
| نویسنده | ||
| S. Shaebani* | ||
| School of Mathematics and Computer Science, Damghan University, P.O. Box 36716-41167, Damghan, Iran. | ||
| چکیده | ||
| A {\it local antimagic labeling} of a connected graph $G$ with at least three vertices, is a bijection $f:E(G) \rightarrow \{1,2,\ldots , |E(G)|\}$ such that for any two adjacent vertices $u$ and $v$ of $G$, the condition $\omega _{f}(u) \neq \omega _{f}(v)$ holds; where $\omega _{f}(u)=\sum _{x\in N(u)} f(xu)$. Assigning $\omega _{f}(u)$ to $u$ for each vertex $u$ in $V(G)$, induces naturally a proper vertex coloring of $G$; and $|f|$ denotes the number of colors appearing in this proper vertex coloring. The {\it local antimagic chromatic number} of $G$, denoted by $\chi _{la}(G)$, is defined as the minimum of $|f|$, where $f$ ranges over all local antimagic labelings of $G$. In this paper, we explicitly construct an infinite class of connected graphs $G$ such that $\chi _{la}(G)$ can be arbitrarily large while $\chi _{la}(G \vee \bar{K_{2}})=3$, where $G \vee \bar{K_{2}}$ is the join graph of $G$ and the complement graph of $K_{2}$. The aforementioned fact leads us to an infinite class of counterexamples to a result of [Local antimagic vertex coloring of a graph, Graphs and Combinatorics 33} (2017), 275-285]. | ||
| کلیدواژهها | ||
| Antimagic labeling؛ Local antimagic labeling؛ Local antimagic chromatic number | ||
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