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Real-world networks evolve over time via additions or removals of nodes and edges. In current network evolution models, node degree varies or grows arbitrarily. A recently introduced degree-preserving network growth (DPG) family of models preserves node degree, resulting in structures significantly different from and more diverse than previous models ([Nature Physics 2021,

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Let 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.