A Lower Bound for Radio k-chromatic Number of an Arbitrary Graph

Radio $k$-coloring is a variation of Hale's channel assignment problem, in which one seeks to assign positive integers to the vertices of a graph $G$, subject to certain constraints involving the distance between the vertices. Specifically, for any simple connected graph $G$ with diameter $d$ and a positive integer $k$, $1\leq k \leq d$, a radio $k$-coloring of $G$ is an assignment $f$ of positive integers to the vertices of $G$ such that $|f(u)-f(v)|\geq 1+k-d(u, v)$, where $u$ and $v$ are any two distinct vertices of $G$ and $d(u, v)$ is the distance between $u$ and $v$. In this paper we give a lower bound for the radio $k$-chromatic number of an arbitrary graph in terms of $k$, the total number of vertices $n$ and a positive integer $M$ such that $d(u,v)+d(v,w)+d(u,w)\leq M$ for all $u,v,w\in V(G)$. If $M$ is the triameter we get a better lower bound. We also find the triameter $M$ for several graphs, and show that the lower bound obtained for these graphs is sharp for the case $k=d$.