Multi-level distance labelings for generalized gear graph

M.T Rahim, M Farooq, Murtaza Ali, S Jan

Abstract


The \emph{radio number} of $G$, $rn(G)$, is the minimum possible span. Let $d(u,v)$ denote the \emph{distance} between two distinct vertices of a connected graph $G$ and $diam(G)$ be the \emph{diameter} of $G$. A \emph{radio labeling} $f$ of $G$ is an assignment of positive integers to the vertices of $G$ satisfying $d(u,v)+|f(u)-f(v)|\geq diam(G)+1$. The largest integer in the range of the labeling is its span. In this paper we show that $rn(J_{t,n})\geq\left\{ \begin{array}{ll} \frac{1}{2}(nt^{2}+2nt+2n+4), & \hbox{when t is even;} \\ \frac{1}{2}(nt^{2}+4nt+3n+4), & \hbox{when t is odd.} \end{array} \right.$\\ \noindent Further the exact value for the radio number of $J_{2,n}$ is calculated.

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