On the Radio $k$-chromatic Number of Paths

A radio $k$-coloring of a graph $G$ is an assignment $f$ of positive integers (colors) to the vertices of $G$ such that for any two vertices $u$ and $v$ of $G$, the difference between their colors is at least $1+k-d(u,v)$. The span $rc_k(f)$ of $f$ is $\max\{f(v):v\in V(G)\}$. The radio $k$-chromatic number $rc_k(G)$ of $G$ is $min\lbrace rc_k(f) : f { is a radio k\text{-}coloring of } G\rbrace$. In this paper, in an attempt to prove a conjecture on the radio $k$-chromatic number of path, we determine the radio $k$-chromatic number of paths $P_n$ for $k+5\leq n\leq\frac{7k-1}{2}$ if $k$ is odd and $k+4\leq n\leq\frac{5k+4}{2}$ if $k$ is even.