AN EDGE PRIME OF SOME GRAPHS

M. Simaringa and K. Santhoshkumar

  DOI:
  https://doi.org/10.37418/amsj.9.4.49

Full Text

Let $G=(V,E)$ be a $(l,m)$ graph. A bijection $g:E\rightarrow\{1,2,…,m\}$ is said to be an edge prime labeling if for each edge $ab\in E$, we have \[gcd(g^+ (a),g^+ (b))=1,\] where $g^+ (a)=\sum_{ac\in E} g(ac)$. Moreover, a bijection $g:E\rightarrow\{1,2,…,m\}$ is semiedge prime labeling if for each $ab\in E$, either $gcd(g^+ (a),g^+ (b) )=1$ or $g^+ (a)=g^+ (b)$. A graph that admits an edge prime (or semiedge prime) labeling is said to be an edge prime (or semiedge prime) graph. In this paper, we prove that if $G$ has an edge prime, then $G\cup P_n$ is an edge prime graph. Also, we obtain $\theta(3^{[m]})\odot\theta(3^{[n]})$, $n{\displaystyle \not \equiv}0 (mod\ 5)$ and some graphs superimposing of path are an edge (or semiedge) prime graph.


Keywords:
Prime labeling, Edge prime labeling, Semiedge prime labeling.