Improved Bounds for Some of the Radio \(k\)-chromatic Numbers of Paths

For a path \( P_n \) of order \( n \) and for any odd integer \( k \), \( 1 \leq k \leq n – 3 \), 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 \( \frac{n-4}{2} \leq k < \frac{2n-5}{3} \) and \( \frac{2n-5}{3} \leq k \leq n-3 \). 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 \( k \), \( 3 \leq k \leq n-3 \).