On the Radio Number of the Hexagonal Mesh

Abstract

A radio labeling of a connected graph $G$ is an injection $f$ from the vertices of $G$ to the natural numbers such that $d(u,v)+|f(u)–f(v)|\ge 1+\mathrm{diam}(G)$ for every pair of distinct vertices $u$ and $v$ of $G$. The radio number of $f$, denoted $rn(f)$, is the maximum number assigned to any vertex of $G$. The radio number of $G$, denoted $rn(G)$, is the minimum value of $rn(f)$ taken over all labelings $f$ of $G$. In this paper, we determine bounds for the radio number of the hexagonal mesh.