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原文摘要

For a simple graph $G$, let $d(u,v)$ denote the distance between two vertices $u$ and $v$ in $G$, and let $diam(G)$ denote the diameter of $G$. A radio labeling of $G$ is a mapping $f : V(G) \rightarrow \mathbb{N} \cup \{0\}$ such that $|f(u)-f(v)|\geq diam(G) + 1 - d(u,v)$ holds for every pair of distinct vertices $u, v \in G$. The span of $f$ is the difference between the largest and smallest integers assigned by $f$, and the radio number $rn(G)$ is the minimum span taking over all radio labelings of $G$. Let $P_n$ denote the path on $n$ vertices and let $H$ denote the Heawood graph (see Figure-1). In this paper we determine the radio number for the Cartesian product of a path and the Heawood graph, and characterize all optimal radio labelings of $P_n \Box H$. Moreover, we show that the radio number of $P_n \Box H$ exceed the lower bound given for $rn(G)$ in [6], where $G$ is any graph, and that this gap is arbitrarily large.