Real-world networks evolve over time via the addition or removal of nodes and edges. Their evolution is often constrained by the cost and the capacity of the network elements. The degree of the node representing the node's capacity to form links is one such dynamical constraint. The degree is simply the number of connections the node has. Maintaining connections bears a cost, due to which the process of creating new connections in physical networks is often accompanied by loss of connection between existing nodes. All current network models largely ignore this mechanism despite the fact that it is prevalent, as we show.We introduce a novel family of network growth models called 'Degree Preserving Network Growth' (DPG), in which new nodes join the network over time without changing the existing node's degrees. It is a simple stylized model that captures degree preservation and its consequences in network evolution. Key aspects of this model are mathematically tractable, which allows us to understand the rich network structures generated by DPG that are significantly different from those produced by existing models.We show that DPG can generate the exact structure of most real-world networks despite the NP-hard complexity of such a problem. Moreover, DPG can grow scale-free networks with arbitrary exponents, used extensively in network studies. It does so without any explicit or implicit preferential bias, unlike in all other popular models. DPG can be used to grow networks with desired higher-order metrics (other than the degrees), allowing applications, for example, in epidemic control via network immunization, viral marketing, knowledge dissemination, and the design of molecular isomers with desired properties. Using DPG, we have also discovered a novel connection between prime numbers and graph theory.