Cartesian powers of graphs and consecutive radio labelings

TL;DR

This paper investigates $k$-radio labelings of graphs, focusing on the case where $k$ equals the graph's diameter, and constructs high-diameter graphs with consecutive integer labelings using Cartesian products.

Contribution

The paper introduces constructions of high-diameter graphs with consecutive radio labelings and analyzes the impact of Cartesian products on such labelings.

Findings

Constructed arbitrarily high diameter graphs with consecutive integer labelings.

Analyzed the effect of Cartesian products on $k$-radio labelings.

Provided new examples of graphs with rare high-diameter properties.

Abstract

For $k\in\mathbb{Z}^+$ and $G$ a simple connected graph, a $k$-radio labeling $f:V_G\to\Z^+$ of $G$ requires all pairs of distinct vertices $u$ and $v$ to satisfy $|f(u)-f(v)|\geq k+1-d(u,v)$. When $k=1$, this requirement gives rise to the familiar labeling known as vertex coloring for which each vertex of a graph is labeled so that adjacent vertices have different "colors". We consider $k$-radio labelings of $G$ when $k=\diam(G)$. In this setting, no two vertices can have the same label, so graphs that have radio labelings of consecutive integers are one extreme on the spectrum of possibilities. Examples of such graphs of high diameter are especially rare and desirable. We construct examples of arbitrarily high diameter, and explore further the tool we used to do this -- the Cartesian product of graphs -- and its effect on radio labeling.