Necklaces and scalability of Kautz digraphs

In this paper, the following results are reported. The notion of Kautz necklaces, similar to de Bruijn ones, is introduced. A linear-time algorithm for generating Kautz necklaces in lexicographic ordering is described and a formula for enumerating the Kautz necklaces is given. A one-to-one mapping between de Bruijn and Kautz vertices preserving the necklaces is presented. Then the notion of necklaces is generalized and a simple necklace-based algorithm for factorization Kautz digraphs is described. Using this factorization, a construction of d-regular partial line digraphs of d-regular Kautz digraphs is described. This result implies that evenly distributed in the space between d-regular Kautz digraphs of diameter D and D+1 there are d-2 digraphs with the same routing and connectivity properties. Finally, a generalization of the method enables to construct partial line digraphs of any order with the connectivity close to d. These results make the Kautz digraphs unique in that sense that they are not only the very dense digraphs, but also very well incrementally scalable