Radio Multiplicative Number of Certain Classes of Transformation Graphs

Abstract

The concept of radio labeling motivated by the channel assignment problem is generalised herein to include various other types of radio labelings. Let $\mathbb{M}$ be a subset of non-negative integers and $(\mathbb{M},\star)$ be a monoid with the identity $e$.
We define a radio $\star$-labeling of graph $G(V,E)$ as a mapping $f:V\rightarrow \mathbb{M}$ such that $| f(u) - f(v)|\star\ d(u,
v)\geq diam(G)+1-e $, for all $u,v\in V$. The radio $\star$-number $rn^\star(f)$ of a radio $\star$-labeling $f$ of $G$ is the maximum label assigned to a vertex of $G$. The \textit{radio $\star$-number} of $G$ denoted by $rn^\star(G)$ is $min\{ rn^\star(f)\} $ taken over all radio $\star$-labeling $f$ of $G$. In this paper we completely determine $rn^\times(G)$ of some transformation graphs of path and cycle.