On Coprime-Preserving Transformations and Dynamic Coprime Labeling
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866915885730496512 |
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| author | Tonapi, Anushka Paquin, Dana |
| author_facet | Tonapi, Anushka Paquin, Dana |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_22552 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On Coprime-Preserving Transformations and Dynamic Coprime Labeling Tonapi, Anushka Paquin, Dana Combinatorics 05C22 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. |
| title | On Coprime-Preserving Transformations and Dynamic Coprime Labeling |
| topic | Combinatorics 05C22 |
| url | https://arxiv.org/abs/2603.22552 |