Uncountably many minimal hereditary classes of graphs of unbounded clique-width

Given an infinite word over the alphabet {0, 1, 2, 3}, we define a class of bipartite hereditary graphs G alpha , and show that G alpha has unbounded clique-width unless alpha contains at most finitely many non-zero letters. We also show that G alpha is minimal of unbounded clique-width if and only if alpha belongs to a precisely defined collection of words Gamma. The set Gamma includes all almost periodic words containing at least one non-zero letter, which both enables us to exhibit uncountably many pairwise distinct minimal classes of unbounded clique width, and also proves one direction of a conjecture due to Collins, Foniok, Korpelainen, Lozin and Zamaraev. Finally, we show that the other direction of the conjecture is false, since Gamma also contains words that are not almost periodic.