radio antipodal colouring; radio number; distance labeling
The radio antipodal number of a graph $G$ is the smallest integer $c$ such that there exists an assignment $f\: V(G)\rightarrow \lbrace 1,2,\ldots ,c\rbrace $ satisfying $|f(u)-f(v)|\ge D-d(u,v)$ for every two distinct vertices $u$ and $v$ of $G$, where $D$ is the diameter of $G$. In this note we determine the exact value of the antipodal number of the path, thus answering the conjecture given in [G. Chartrand, D. Erwin and P. Zhang, Math. Bohem. 127 (2002), 57–69]. We also show the connections between this colouring and radio labelings.
 G. Chartrand, D. Erwin, F. Harary, P. Zhang: Radio labelings of graphs
. Bull. Inst. Combin. Appl. 33 (2001), 77–85. MR 1913399
 G. Chartrand, D. Erwin, P. Zhang: Radio antipodal colorings of cycles
. Congr. Numerantium 144 (2000), 129–141. MR 1817928
 G. Chartrand, D. Erwin, P. Zhang: Radio antipodal colorings of graphs
. Math. Bohem. 127 (2002), 57–69. MR 1895247
 D. Liu, X. Zhu: Multi-level distance labelings for paths and cycles. (to appear).