New c-ary Perfect Factors in the de Bruijn Graph

A c-ary Perfect Factor is a collection of uniformly long cycles whose elements are drawn from a set of size c, in which every possible v-tuple of elements occurs exactly once. In the binary case, i.e. where c = 2, these perfect factors have previously been studied by Etzion, [1], who showed that the necessary conditions for their existence are in fact sufficient. This result has recently been extended by Paterson, [2], who has shown that the necessary existence conditions are sufficient whenever c is a prime power. In [3] the existence question for general composite c was studied, and it was conjectured that the necessary existence conditions are also sufficient in this case. However, although construction methods for these Perfect Factors were exhibited in [3], the conjecture remains open. In this paper we provide further evidence for the conjecture by constructing c-ary Perfect Factors for several of the previously undecided cases.