Radio antipodal colorings of graphs

Abstract

A radio antipodal coloring of a connected graph $G$ with diameter $d$ is an assignment of positive integers to the vertices of $G$, with $x \in V(G)$ assigned $c(x)$, such that \[ d(u, v) + |c(u) -c(v)| \ge d \] for every two distinct vertices $u$, $v$ of $G$, where $d(u, v)$ is the distance between $u$ and $v$ in $G$. The radio antipodal coloring number $\mathop {\mathrm ac}(c)$ of a radio antipodal coloring $c$ of $G$ is the maximum color assigned to a vertex of $G$. The radio antipodal chromatic number $\mathop {\mathrm ac}(G)$ of $G$ is $\min \lbrace \mathop {\mathrm ac}(c)\rbrace $ over all radio antipodal colorings $c$ of $G$. Radio antipodal chromatic numbers of paths are discussed and upper and lower bounds are presented. Furthermore, upper and lower bounds for radio antipodal chromatic numbers of graphs are given in terms of their diameter and other invariants.