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On Coprime-Preserving Transformations and Dynamic Coprime Labeling

Anushka Tonapi, Dana Paquin

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.

On Coprime-Preserving Transformations and Dynamic Coprime Labeling

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 -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.
Paper Structure (5 sections, 10 theorems, 47 equations)

This paper contains 5 sections, 10 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Preliminaries: Classical Prime and Coprime Labelings
  3. Definitions and Notation
  4. Main Results
  5. Example: $n=561$

Key Result

Theorem 4.1

A finite graph $G$ admits a dynamic coprime labeling (with respect to some coprime-preserving map $g$) if and only if it admits a classical coprime labeling.

Theorems & Definitions (32)

  • Definition 2.1: Prime labeling
  • Definition 2.2: Coprime labeling
  • Definition 3.1: Coprime-preserving transformation
  • Definition 3.2: Dynamic coprime labeling (DCL)
  • Definition 3.3: Periodic DCL
  • Definition 3.4: $n$--hypercube graph
  • Definition 3.5: Bounded/Unbounded Labelings
  • Remark 3.6
  • Theorem 4.1: Existence Equivalence
  • proof
  • ...and 22 more