Colouring of cycles in the de Bruijn graphs

We show that the problem of finding the family of all so called the locally reducible factors in the binary de Bruijn graph of order k is equivalent to the problem of finding all colourings of edges in the binary de Bruijn graph of order k-1, where each vertex belongs to exactly two cycles of different colours. In this paper we define and study such colouring for the greater class of the de Bruijn graphs in order to define a class of so called regular factors, which is not so difficult to construct. Next we prove that each locally reducible factor of the binary de Bruijn graph is a subgraph of a certain regular factor in the m-ary de Bruijn graph.