# Elegant vertex labelings with prime numbers

## TL;DR

This paper explores a novel graph labeling method assigning odd primes to vertices, aiming to produce edge differences that cover the first even numbers, with a conjecture about paths and an algorithm for constructing such labelings.

## Contribution

It introduces the concept of elegant prime labelings for graphs, especially trees and paths, and provides an algorithm to generate elegant paths for up to 3500 vertices.

## Key findings

- Successfully generated elegant paths for all n up to 3500
- Conjecture that all paths are elegant with prime labelings
- Proposes an algorithm for constructing elegant labelings

## Abstract

We consider graph labelings with an assignment of odd prime numbers to the vertices. Similarly to graceful graphs, a labeling is said to be elegant if the absolute differences between the labels of adjacent vertices describe exactly the first even numbers. The labels of an elegant tree with $n$ vertices are the first $n$ odd prime numbers and we want that the resulting edge labels are exactly the first even numbers up to $2n-2$. We conjecture that each path is elegant and we give the algorithm with which we got elegant paths of $n$ primes for all $n$ up to $n=3500$.