# A different approach for multi-level distance labellings of path structure networks

## Main Article Content

## Abstract

For a positive integer $k$, a radio $k$-labelling of a simple connected graph $G=(V, E)$ is a mapping $f$ from the vertex set $V(G)$ to a set of non-negative integers such that $|f(u)-f(v)|\geqslant k+1-d(u,v)$ for each pair of distinct vertices $u$ and $v$ of $G$, where $d(u,v)$ is the distance between $u$ and $v$ in $G$. The \emph{span} of a radio $k$-coloring $f$, denoted by $span_f(G)$, is defined as $\displaystyle\max_{v\in V(G)}f(v)$ and the \emph{radio $k$-chromatic number of $G$}, denoted by $rc_k(G)$, is $\displaystyle\min_{f}\{~span_f(G)\}$ where the minimum is taken over all radio $k$-labellings of $G$. In this article, we present results of radio $k$-chromatic number of path $P_n$ for $k\in\{n-1, n-2,n-3\}$ in different approach but simple way.

## Article Details

*Tamkang Journal of Mathematics*. https://doi.org/10.5556/j.tkjm.55.2024.3913

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.

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