Abstract
AbstractLet us call a simple graph on $$n\geqslant 2$$
n
⩾
2
vertices a prime gap graph if its vertex degrees are 1 and the first $$n-1$$
n
-
1
prime gaps. We show that such a graph exists for every large n, and in fact for every $$n\geqslant 2$$
n
⩾
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.
Funder
National Research, Development and Innovation Office
National Science Foundation
Publisher
Springer Science and Business Media LLC
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