Long paths and cycles in hypercubes with faulty vertices

Long cycles in hypercubes with optimal number of faulty vertices

“…Therefore, for every dimension j there exists exactly one other dimension k such that e j,k / ∈ F , so all dimensions are split into three pairs {j 1 , k 1 }, {j 2 , k 2 } and {j 3 , k 3 } such that e j 1 ,k 1 , e j 2 ,k 2 , e j 3 ,k 3 / ∈ F . This is satisfied up to isomorphism only by one set of faulty vertices F : the set of all vertices of level 0 or 2 except the vertices e 1,2 , e 3,4 and e 5,6 . By Lemma 5.1 with the assumption (i), it suffices to find a long F 6:L -free cycle in Q 6:L which is presented on Figure 2.…”

Paired 2-disjoint path covers and strongly Hamiltonian laceability of bipartite hypercube-like graphs

“…A path or cycle in a graph is called Hamiltonian if it contains all the vertices of the graph. This problem has attracted much attention in the literature, such as works on faulty hypercubes [9,10,17]. The disjoint path cover problem is closely related to the Hamiltonian problem in that a Hamiltonian path joining a pair of vertices can be viewed as any type of 1-DPC joining them, and a Hamiltonian path joining a pair of vertices that passes through k − 1 prescribed edges can be obtained directly from some paired k-DPC of the graph [29].…”

Edge-fault-tolerant diameter and bipanconnectivity of hypercubes

“…For (fault-tolerant) path and/or cycle embedding into hypercubes, see recent Refs. [2][3][4][5][6][7][8][9][10][11][13][14][15]17,[19][20][21][22] and a survey [23].…”