Cycles embedding on folded hypercubes with faulty nodes

Every edge lies on cycles embedding in folded hypercubes with vertex-fault-tolerant

“…In this paper, we extend Cheng's [2] results to embedding more cycles on FQ n with faulty vertex f . We obtain the following two properties:…”

“…After that Xu [25] extended the above result to show that every fault-free edge of FQ n − F F e lies on a cycle of every even length from 4 to 2 n ; if n is even, every edge of FQ n − F F e also lies on a cycle of every odd length from n + 1 to 2 n − 1, where |F F e | ≤ n − 1. Recently, Cheng [2] showed that every fault-free edge of FQ n − F F v lies on a cycle of every even length from 4 to 2 n − 2|F F v | if n ≥ 3, and if n ≥ 2 is even, every edge of FQ n − F F v also lies on a cycle of every odd length from n + 1 to…”

Odd cycles embedding on folded hypercubes with conditional faulty edges

“…Lemma 3 [5]. Assume that FQ n is partitioned along dimension i (1 6 i 6 n) into two ðn À 1Þ-cubes, denoted by Q 0 nÀ1 and Q 1 nÀ1 , and 0u and 0v (respectively, 1u and 1v) are two nodes in Q 0 nÀ1 (respectively, Q Let F v (respectively, F e ) be the set of faulty vertices (respectively, edges) in Q n .…”

“…Hsieh et al [18] considered FQ n with only one faulty vertex f and showed that FQ n À ff g contains a fault-free cycle of every even length from 4 to 2 n À 2 if n P 3 and every odd length from n þ 1 to 2 n À 1 if n P 2 is even. Cheng et al [5] studied FQ n with jFF v j 6 n À 2 faulty vertices and showed that if n P 3, then every edge of FQ n À FF v lies on a fault-free cycle of every even length from 4 to 2 n À 2jFF v j; and if n P 2 is even, then every edge of FQ n À FF v lies on a fault-free cycle of every odd length from n þ 1 to 2 n À 2jFF v j À 1. Hsieh [15] considered the FQ n with both faulty edges and faulty vertices and proved that there exists a fault-free cycle of length at least 2 n À 2jFF v j in FQ n if jF v j þ jF e j 6 n À 1, where n P 3.…”

Edge-pancyclicity and edge-bipancyclicity of faulty folded hypercubes

“…Hence, the issue of fault-tolerant cycle embedding in an n-dimensional folded hypercube F Q n has been studied in [2,5,7,8,9,10,11,12,13,14,20,19]. Embedding cycles in networks is important as many network algorithms utilize cycles as data structure.…”