Consecutive Radio Labeling of Hamming Graphs
Published in arXiv, 2020
For a graph \(G\), a \(k\)-radio labeling of \(G\) is the assignment of positive integers to the vertices of \(G\) such that the closer two vertices are on the graph, the greater the difference in labels is required to be. Specifically, \(\vert f(u)−f(v)\vert\geq k+1−d(u,v)\) where \(f(u)\) is the label on a vertex \(u\) in \(G\). Here, we consider the case when \(G\) is the Cartesian products of complete graphs. Specifically we wish to find optimal labelings that use consecutive integers and determine when this is possible. We build off of a paper by Amanda Niedzialomski and construct a framework for discovering consecutive radio labelings for Hamming Graphs, starting with the smallest unknown graph, $$K_3^4, for which we provide an optimal labeling using our construction.