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arXiv · 2603.22552

On Coprime-Preserving Transformations and Dynamic Coprime Labeling

Anushka Tonapi·Dana Paquin
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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.

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Anushka Tonapi, Dana Paquin. 2026-03-23. On Coprime-Preserving Transformations and Dynamic Coprime Labeling. https://arxiv.org/abs/2603.22552

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