A Meyniel obstruction is an odd cycle with at most one chord and a graph is Meyniel if and only if it has no Meyniel obstruction. The authors have found in a previous work a linear-time algorithm for optimaly color the vertices of a Meyniel graph. Here we extend this algorithm to obtain a linear-time algorithm that, for any graph, colors optimally the vertices of the graph or finds a Meyniel obstruction.