The sequence of prime gaps is graphic

TL;DR

This paper demonstrates that for every large enough number, there exists a simple graph whose degrees correspond to the first few prime gaps, establishing a novel connection between prime number theory and graph theory.

Contribution

It proves the existence of prime gap graphs for all large n and under the Riemann hypothesis for all n ≥ 2, introducing an infinite sequence of such graphs generated by a degree preserving process.

Findings

Prime gap graphs exist for all sufficiently large n.

An infinite sequence of prime gap graphs can be generated systematically.

First identification of a naturally occurring infinite sequence of positive integers as graphic.

Abstract

Let us call a simple graph on $n\geq 2$ vertices a prime gap graph if its vertex degrees are $1$ and the first $n-1$ prime gaps. We show that such a graph exists for every large $n$, and in fact for every $n\geq 2$ if we assume the Riemann hypothesis. Moreover, an infinite sequence of prime gap graphs can be generated by the so-called degree preserving growth process. This is the first time a naturally occurring infinite sequence of positive integers is identified as graphic. That is, we show the existence of an interesting, and so far unique, infinite combinatorial object.