OPTIMAL RADIOCOLORING OF TREES

Abstract

A Radiocoloring (RC) of a graph $G$ is a function $f$ from the vertex set $V(G)$ to the set of all non-negative integers (labels) such that $|f(u)-f(v)|\geq 2$ if $d(u,v) = 1$ and $|f(u)-f(v)|\geq 1$ if $d(u, v) = 2$. The number of discrete labels and the range of labels used are called order and span, respectively. In this paper, we concentrate on the minimum order span Radiocoloring Problem (RCP) of trees. The optimization version of the minimum order span RCP that tries to find, from all minimum order assignments, one that uses the minimum span. We provide attainable lower and upper bounds for trees. Moreover, a complete characterization of caterpillars (as a subclass of trees) with the minimum order span is given.