Radio Number for the Cartesian Product of Two Trees

TL;DR

This paper establishes bounds and conditions for the radio number of the Cartesian product of two trees, including specific cases like two stars and a path with a star, advancing understanding of graph labeling.

Contribution

It provides a lower bound for the radio number of the Cartesian product of two trees and characterizes when this bound is achieved, including exact values for specific tree products.

Findings

Lower bound for the radio number of the Cartesian product of two trees.

Necessary and sufficient conditions for achieving the bound.

Exact radio number for the product of two stars and a path with a star.

Abstract

Let $G$ be a simple connected graph. For any two vertices $u$ and $v$, let $d(u,v)$ denote the distance between $u$ and $v$ in $G$, and let $diam(G)$ denote the diameter of $G$. A radio-labeling of $G$ is a function $f$ which assigns to each vertex a non-negative integer (label) such that for every distinct vertices $u$ and $v$ in $G$, it holds that $|f(u)-f(v)| \geq diam(G) - d(u,v) +1$. The span of $f$ is the difference between the largest and smallest labels of $f(V)$. The radio number of $G$, denoted by $rn(G)$, is the smallest span of a radio labeling admitted by $G$. In this paper, we give a lower bound for the radio number of the Cartesian product of two trees. Moreover, we present three necessary and sufficient conditions, and three sufficient conditions for the product of two trees to achieve this bound. Applying these results, we determine the radio number of the Cartesian product of two stars as well as a path and a star.