Radio number of trees

TL;DR

This paper investigates the radio number of trees, establishing conditions for achieving bounds and determining the radio number for specific tree families, advancing understanding of graph labelings.

Contribution

It provides necessary and sufficient conditions for lower bounds and an upper bound on the radio number of trees, and calculates the radio number for three tree families.

Findings

Established conditions for bounds on the radio number of trees.

Derived an upper bound on the radio number of trees.

Determined the radio number for three specific families of trees.

Abstract

A radio labeling of a graph $G$ is a mapping $f: V(G) \rightarrow \{0, 1, 2, \ldots\}$ such that $|f(u)-f(v)|\geq d + 1 - d(u,v)$ for every pair of distinct vertices $u, v$ of $G$, where $d$ is the diameter of $G$ and $d(u,v)$ the distance between $u$ and $v$ in $G$. The radio number of $G$ is the smallest integer $k$ such that $G$ has a radio labeling $f$ with $\max\{f(v) : v \in V(G)\} = k$. We give a necessary and sufficient condition for a lower bound on the radio number of trees to be achieved, two other sufficient conditions for the same bound to be achieved by a tree, and an upper bound on the radio number of trees. Using these, we determine the radio number for three families of trees.