Determining the Radio Number for the Cartesian Product of a Tree and a Complete Graph

Core Concepts

The paper provides a tight lower bound for the radio number of the Cartesian product of a tree and a complete graph, and gives necessary and sufficient conditions as well as sufficient conditions to achieve this lower bound. The radio number for the Cartesian product of a level-wise regular tree and a complete graph is also determined.

Abstract

The paper focuses on the radio labeling problem for the Cartesian product of a tree and a complete graph. The key highlights and insights are:

The paper provides a lower bound for the radio number of the Cartesian product of a tree and a complete graph (Theorem 3.1).

It gives two necessary and sufficient conditions (Theorems 3.2 and 3.3) and three sufficient conditions (Theorem 3.4) to achieve the lower bound.

Using these results, the radio number for the Cartesian product of a level-wise regular tree and a complete graph is determined (Theorem 4.1).

The radio number for the Cartesian product of a path and a complete graph, derived in prior work, can be obtained as a corollary of the results presented in this paper.

The paper introduces key concepts related to radio labeling, such as weight centers of a tree, branches, and level-wise regular trees, and leverages these to derive the main results.

The proofs rely on carefully analyzing the structure of the Cartesian product graph and the properties of the optimal radio labeling.

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Key Insights Distilled From

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

by Payal Vasoya... at arxiv.org 04-15-2024

https://arxiv.org/pdf/2404.08400.pdf

Deeper Inquiries

Can the results be extended to other graph products beyond the Cartesian product

The results obtained for the Cartesian product of a tree and a complete graph can potentially be extended to other graph products. The key lies in understanding the structural properties of the graphs involved in the product and how they interact with each other. By analyzing the characteristics of different graph products and their impact on radio labeling, it may be possible to generalize the findings to other types of graph products. However, the specific details and considerations would vary depending on the nature of the graph products being studied.

Are there efficient algorithms to construct the optimal radio labeling for the Cartesian product of a tree and a complete graph

Constructing the optimal radio labeling for the Cartesian product of a tree and a complete graph can be approached using algorithmic techniques. One possible approach is to adapt existing algorithms for radio labeling on individual graphs and extend them to handle the Cartesian product. This may involve considering the unique properties of trees and complete graphs, as well as how they combine in the Cartesian product. By developing efficient algorithms that take into account the specific characteristics of the Cartesian product, it is possible to construct the optimal radio labeling in a systematic and effective manner.

What are the practical applications of determining the radio number for the Cartesian product of a tree and a complete graph

Determining the radio number for the Cartesian product of a tree and a complete graph has practical applications in various fields. One such application is in wireless communication networks, where channel assignment plays a crucial role in minimizing interference and optimizing network performance. By understanding the radio number for this specific graph product, network planners can make informed decisions about channel assignments and optimize the overall network efficiency. Additionally, the results can be applied in areas such as sensor networks, distributed computing, and telecommunications, where efficient channel allocation is essential for smooth operation and resource utilization.