Minimal Eulerian trail in a labeled digraph

Let $G$ be an Eulerian directed graph with an arc-labeling such that arcs going out from the same vertex have different labels. In this work, we present an algorithm to construct the Eulerian trail starting at an arbitrary vertex $v$ of minimum lexicographical label among labels of all Eulerian trails starting at this vertex. We also show an application of this algorithm to construct the minimal de Bruijn sequence of a language.