On Coprime-Preserving Transformations and Dynamic Coprime Labeling

TL;DR

This paper introduces dynamic coprime labeling (DCL), a new framework for maintaining coprimality in evolving graphs, proving its equivalence to classical coprime labeling and exploring its applications.

Contribution

It extends coprime labeling to dynamic, time-sensitive networks, characterizes coprime-preserving transformations, and proves DCL's existence for various graph families.

Findings

DCL exists if and only if classical coprime labeling exists.

DCL applies to paths, wheels, cycles, and hypercubes.

Introduces coprime-preserving transformations and applies DCL to Carmichael's theorem.

Abstract

In this paper, we introduce dynamic coprime labeling (DCL), a novel extension of coprime labeling for time-sensitive networks. In particular, we explore whether there exists a graph labeling scheme that maintains relative coprimality among adjacent vertices as the graph evolves over time. We extend the definition of coprime labeling to include an injective labeling function, a time variable, and a transformation function. A DCL on a finite simple graph is a sequence of injective vertex labelings with the property that every edge is labeled by coprime integers at each time step, and the evolution is given by a time-independent coprime-preserving transformation. We prove that a graph admits a DCL if and only if it admits a classical coprime labeling (existence equivalence). We characterize families of coprime-preserving transformations and provide proofs of the existence of DCLs for paths, wheels, cycles, and the $n$-hypercube. We also introduce two classes of coprime-preserving transformations and present an application of DCL to Carmichael's theorem. These results establish DCL as a rigorous framework for further algorithmic and applied investigations.