De Bruijn Graphs and their Applications to Fault Tolerant Networks

28 JOEL BAKERFigure 17. UB(2, 3) (left) and the modification (right).of a single node failure on B(2, n) that we could use the technique of removing a necklaceto show that these graphs admit cycles of length greater than or equal to 2 n − n − 1.We then saw in Section 4.2 that the standard De Bruijn graph B(d, n) admits at leastone cycle of length d n − 1 under d − 1 edge failures when d is a prime power. In orderto prove this bound we used irreducible polynomials and recurrence relations in order toconstruct these cycles. We then saw from [RB93a] that we could use this knowledge toprovide a lower bound for the number of Hamiltonian cycles on B(d, n) for any d, basedon its prime decomposition.Lastly, we examined some modifications to De Bruijn graphs and sequences. The Q ncycles gave us an example of a different type of structure that we could examine using thetechniques developed in Section 3. Finally, we looked at the modifications presented byRowley and Bose in [RB91] and [RB93b] that yield slightly stronger results but lost someof the original structure.6.1. Further Directions. There are a few different directions that one could take forfurther research on De Bruijn graphs and their applications. First, we note that the methoddiscussed in Section 4.2 for finding edge disjoint Hamiltonian cycles on De Bruijn graphsis dependent on the fact that our alphabet was chosen with d elements, where d is a primeor prime power. If we choose d to be composite then the current bounds have room forimprovement. Z. Kása proposes in [Kás10] that the number of edge disjoint Hamiltoniancycles is d − 1. This bound differs greatly from the one proposed by Rowley and Bose in[RB93a] discussed in Section 4.2.Another direction one could take is to expand the study of node failure to De Bruijngraphs other than binary De Bruijn graphs. Our investigation in Section 4.1 made use ofthe concept of weight which is not immediately applicable to De Bruijn graphs, B(d, n),where d > 2.