Consecutive Radio Labeling of Hamming Graphs

TL;DR

This paper develops a framework for finding optimal consecutive radio labelings of Hamming graphs, specifically Cartesian products of complete graphs, and demonstrates its effectiveness on the smallest unknown case, $K_3^4$.

Contribution

It introduces a new framework for discovering consecutive radio labelings of Hamming graphs, including the first optimal labeling for $K_3^4$.

Findings

Constructed an optimal labeling for $K_3^4$.

Established a general framework for Hamming graphs.

Extended previous work by Amanda Niedzialomski.

Abstract

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.