Elegant vertex labelings with prime numbers

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 $n=3500$.