ON THE $S_3$-MAGIC GRAPHS

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Abstract

Let $G = (V(G),E(G))$ be a finite $(p, q)$ graph and let $(A,\ast)$ be a finite non-abelain group with identity element $1$. Let $f : E(G) \rightarrow N_{q}=\{1,2,\ldots,q\}$ and let $g : E(G) \rightarrow A \setminus \{1\}$ be two edge labelings of $G$ such that $f$ is bijective. Using these two labelings $f$ and $g$ we can define another edge labeling

$\ell: E(G) \rightarrow N_{q} \times A\setminus \{1\}$ by $$ \ell(e) := (f(e),g(e)) \ \ \text{for all} \ e \in E(G).$$

Define a relation $\leq$ on the range of $\ell$ by:

$$ (f(e),g(e)) \le (f(e'), g(e')) \ \ \text{if and only if } \ \ f(e)\le f(e').$$ This relation $\le $ is a partial order on the range of $\ell$. Let $$\{(f(e_1), g(e_1)) , (f(e_2), g(e_2)), \ldots, (f(e_k), g(e_k))\}$$ be a chain in the range of $\ell$. We define a product of the elements of this chain as follows:

$$\prod_{i=1}^k(f(e_i), g(e_i)) := ((((g(e_1) \ast g(e_2))\ast g(e_3))\ast \cdots)\ast g(e_k) .$$

Let $u \in V$ and let $N^{*}(u)$ be the set of all edges incident with $u$. Note that the restriction of $\ell$ on ${N^{*}(u)}$ is a chain, say $(f(e_1), g(e_1)) \leq (f(e_2), g(e_2)) \leq \cdots \leq (f(e_n),g(e_n))$. We define

\begin{equation*}

\ell^{*} (u):= \prod_{i=1}^n(f(e_i), g(e_i)).

\end{equation*}

If $\ell^{*}(u)$ is a constant, say $a$ for all $u \in V(G)$, we say that the graph $G$ is $A$ - magic. The map $\ell^*$ is called an $A$ -magic labeling of $G$ and the corresponding constant $a$ is called the magic constant. In this paper, we consider the permutation group $S_3$

and investigate graphs that are $S_3$-magic.

Keywords and Phrases

A-magic labeling, non-abelian group, symmetric group $S_3$, $S_3$-magic labeling.

A.M.S. subject classification

05C25, 05C78.

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