2024/04/12 by Payal Vasoya, Vasoya, Payal, Devsi Bantva +1
Computer Science · Engineering · #05C12 #05C15 #05C78 #Advanced Graph Theory Research #Combinatorics (math.CO) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #Graph Labeling and Dimension Problems #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2404.08400
openalex publication_date 2024/04/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A radio labelling of a graph G is a mapping f : V(G) → \0, 1, 2,…\ such that |f(u)-f(v)|≥ 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 ∈ 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.