Bounds of the Radio Number of Stacked-Book Graphs with Odd Paths

TL;DR

This paper establishes tight bounds for the radio number of stacked-book graphs formed from a star and an odd path, extending previous work on even paths and contributing to graph labeling theory.

Contribution

It provides the first tight bounds for the radio number of stacked-book graphs with odd paths, complementing existing results for even paths.

Findings

Derived tight upper and lower bounds for the radio number

Extended the analysis to graphs with odd path lengths

Complemented previous work on even path cases

Abstract

A Stacked-book graph Gm,n is obtained from the Cartesian product of a star graph Sm and a path Pn, where m and s are the orders of the star graph and the path respectively. Obtaining the radio number of a graph is a rigorous process, which is dependent on the diameter of G and positive difference of non-negative integer labels f(u) and f(v) assigned to any two u, v in the vertex set V (G) of G. This paper obtains tight upper and lower bounds of the radio number of Gm,n where the path Pn has an odd order. The case where Pn has an even order has been investigated.