The Radio Antipodal and Radio Numbers of the Hypercube

Abstract

A radio $k$-labeling of a connected graph $G$ is an assignment $f$ of non-negative integers to the vertices of $G$ such that

$$|f(x)–f(y)|\ge k+1–d(x,y),$$

for any two vertices $x$ and $y$, where $d(x,y)$ is the distance between $x$ and $y$ in $G$. The radio antipodal number is the minimum span of a radio $(diam(G)–1)$-labeling of $G$ and the radio number is the minimum span of a radio $(diam(G))$-labeling of $G$.

In this paper, the radio antipodal number and the radio number of the hypercube are determined by using a generalization of binary Gray codes.