Around 1950, Zykov and Blanche Descartes proposed the two first explicit constructions of triangle-free graphs with arbitrarily large chromatic number. We define a Zykov (resp. Blanche Descartes) graph as any induced subgraph of a graph created using Zykov’s construction (resp. Blanche Descartes’s). We give a structural characterization of Zykov graphs based on a specific type of stable set, that we call splitting stable set. It implies that recognizing this class is NP-complete, while being FPT in the treewidth of the input graph. We provide similar results for Blanche Descartes graphs. We also observe that any Blanche Descartes graph is a Zykov graph.