This paper is concerned with the problem of supervised learning of deterministic finite state automata, in the technical sense of identification in the limit from complete data, by finding a minimal DFA consistent with the data (regular inference).We solve this problem by translating it in its entirety to a vertex coloring problem. Essentially, such a problem consists of two types of constraints that restrict the hypothesis space: inequality and equality constraints.Inequality constraints translate to the vertex coloring problem in a very natural way. Equality constraints however greatly complicate the translation to vertex coloring. In previous coloring-based translations, these were therefore encoded either dynamically by modifying the vertex coloring instance on-the-fly, or by encoding them as satisfiability problems. We provide the first translation that encodes both types of constraints together in a pure vertex coloring instance. This offers many opportunities for applying insights from combinatorial optimization and graph theory to regular inference. We immediately obtain new complexity bounds, as well as a family of new learning algorithms which can be used to obtain both exact hypotheses, as well as fast approximations.