Radio number for the Cartesian product of a tree and a complete graph

TL;DR

This work establishes a tight lower bound for the radio number of the Cartesian product $T \Box K_n$ where $T$ is a tree, and develops structural criteria for when the bound is attained. It introduces a framework based on level assignments, weight centers, and distance decompositions to characterize optimal radio labelings and provides two necessary-and-sufficient conditions along with three sufficient conditions to achieve equality. The authors further apply the bound to level-wise regular trees, obtaining explicit closed-form expressions for $rn(T^z \Box K_n)$ (for $z=1,2$) and constructing labelings that attain these values, unifying and extending known results such as the radio number for paths. The results yield constructive approaches for channel assignment problems modeled by $T \Box K_n$, illustrating how tree geometry governs the labeling span and enabling direct computation of optimal radio numbers in these product graphs.

Abstract

A radio labelling of a graph $G$ is a mapping $f : V(G) \rightarrow \{0, 1, 2,\ldots\}$ such that $|f(u)-f(v)|\geq diam(G) + 1 - d(u,v)$ for every pair of distinct vertices $u,v$ of $G$, where $diam(G)$ is the diameter of $G$ and $d(u,v)$ is the distance between $u$ and $v$ in $G$. The radio number $rn(G)$ of $G$ is the smallest integer $k$ such that $G$ admits a radio labelling $f$ with $\max\{f(v):v \in V(G)\} = k$. In this paper, we give a lower bound for the radio number of the Cartesian product of a tree and a complete graph and give two necessary and sufficient conditions to achieve the lower bound. We also give three sufficient conditions to achieve the lower bound. We determine the radio number for the Cartesian product of a level-wise regular trees and a complete graph which attains the lower bound. The radio number for the Cartesian product of a path and a complete graph derived in [Radio number for the product of a path and a complete graph, J. Comb. Optim., 30 (2015), 139-149] can be obtained using our results in a short way.

Theorems & Definitions (19)

• Definition 1.1
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3: Bantva2
• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Theorem 3.3
• proof