On Radio Labeling Edge-Balanced Caterpillars

Abstract

A radio labeling of a simple connected graph $G$ is a function $f:V(G)\to {\mathbb{Z}}^{+}$ such that for every two distinct vertices $u$ and $v$ of $G$,
$$distance(u,v)+|f(u)–f(v)|\ge 1+diameter(G).$$
The radio number of a graph $G$ is the smallest integer $M$ for which there exists a labeling $f$ with $f(v)\le M$ for all $v\in V(G)$. An edge-balanced caterpillar graph is a caterpillar graph that has an edge such that removing this edge results in two components with an equal number of vertices. In this paper, we determine the radio number of particular edge-balanced caterpillars as well as improve the lower bounds of the radio number of other edge-balanced caterpillars.