The sequence of prime gaps is graphic

Abstract

Let us call a simple graph on $n\ge 2$ 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 $n\ge 2$ 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.

14 pages, LaTeX2e; v2: revised version incorporating suggestions by the referee (e.g. the formal remarks below Theorems 2.2 and 2.4 are new)

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