On the Radio Number of the Hexagonal Mesh

A radio labeling of a connected graph \( G \) is an injection \( f \) from the vertices of \( G \) to the natural numbers such that \( d(u, v) + |f(u) – f(v)| \geq 1 + \operatorname{diam}(G) \) 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.