Improved Bounds for Some of the Radio $k$-chromatic Numbers of Paths

Abstract

For a path $P_n$ of order $n$ and for any odd integer $k$, $1\le k\le n–3$, Chartrand et al. have given an upper bound for the radio $k$-chromatic number of $P_n$ as $\frac{k^2+2k+1}{2}$. Here we improve this bound for $\frac{n-4}{2}\le k<\frac{2n-5}{3}$ and $\frac{2n-5}{3}\le k\le n-3$. They are $\frac{k^2+k+4}{2}$ and $\frac{k^2+k+2}{2}$, respectively. Also, we improve the lower bound of Kchikech et al. from $\frac{k^2+3}{2}$ to $\frac{k^2+5}{2}$ for odd integer $k$, $3\le k\le n-3$.