De Bruijn Graphs and their Applications to Fault Tolerant Networks

DE BRUIJN GRAPHS AND THEIR APPLICATIONS TO FAULT TOLERANT NETWORKS 25Theorem 5. The De Bruijn graph B(d, n) with at most d − 1 faulty edges admits a cycleof length d n − 1 when d is a prime power.Proof. We know that every maximal cycle is a cycle of length d n −1 and also that there ared edge disjoint cycles that we can generate in this manner. This means that these cyclescoverd(d n − 1) = d n+1 − dedges. We also notice that none of them use the loop edges since then they would visit anode twice.Additionally, we know from Procedure 1 in Section 2 that there are d n+1 total edges andd loop edges, thus these maximal cycles partition all the d n+1 − d edges which are not loopedges. This means that since we have d edge disjoint cycles, that removing d−1 edges fromthe whole graph cannot break all of these cycles and therefore when d is a prime power wesee that B(d, n) still admits at least one of these cycles.□Rowley and Bose expand this result to provide a lower bound for De Bruijn graphs whichdo not have alphabets with prime power sizes in [RB93a] . They prove by similar meansin the following Lemmas and Theorem.Lemma 6. The graph B(d, n) admits d − 1 disjoint Hamiltonian cycles when d is a powerof 2.Lemma 7. The graph B(d, n) admits d−12disjoint Hamiltonian cycles when d is an oddprime power.We notice that these differ from our previous result in that they are full Hamiltoniancycles and not just cycles of length d n −1 as before. It is this small difference which reducesthe disjoint cycle count.Lemma 8. If B(d, n) admits k disjoint Hamiltonian cycles, B(d ′ , n) admits k ′ Hamiltoniancycles and d and d ′ are relatively prime, then B(dd ′ , n) admits at least kk ′ Hamiltoniancycles.Theorem 6. The number of disjoint Hamiltonian cycles in B(d, n) is at leastk∏2 −k e(p i i − 1)i=1where p e 11 · · · pe kkis the prime factorization of d.Proof. The proof of this theorem follows directly from Lemmas 6, 7 and 8.□