Hamiltonian Cycles with Prescribed Edges in Hypercubes

Connectivity of Matching Graph of Hypercube

“…We prove that there exists δ ∈ [3] such that c(P ∪ I δ 3 ) = 1. Suppose on the contrary that c(P ∪ I δ 3 ) = 3 for every δ ∈ [3]. The graph on edges P ∪ I δ 3 consists of two common edges and one cycle of size 4.…”

“…Let P be an odd perfect matching of B(Q 3 ). Therefore, c(P ∪ I δ 3 ) is 1 or 3 for every δ ∈ [3]. If there exists δ ∈ [3] such that c(P ∪ I δ 3 ) = 1, then we choose R := Y := I δ 3 and X ∈ Γ(R).…”

“…Therefore, c(P ∪ I δ 3 ) is 1 or 3 for every δ ∈ [3]. If there exists δ ∈ [3] such that c(P ∪ I δ 3 ) = 1, then we choose R := Y := I δ 3 and X ∈ Γ(R). We prove that there exists δ ∈ [3] such that c(P ∪ I δ 3 ) = 1.…”

“…If there exists δ ∈ [3] such that c(P ∪ I δ 3 ) = 1, then we choose R := Y := I δ 3 and X ∈ Γ(R). We prove that there exists δ ∈ [3] such that c(P ∪ I δ 3 ) = 1. Suppose on the contrary that c(P ∪ I δ 3 ) = 3 for every δ ∈ [3].…”

“…An interested reader can find more details about this topic in the survey of Savage [10]. Dvořák [3] showed that every set of at most 2d − 3 edges of Q d (d ≥ 2) that induces vertex-disjoint paths is contained in a Hamiltonian cycle. Dimitrov et al [1] proved that for every perfect matching P of Q d (d ≥ 3) there exists some Hamiltonian cycle that faults P if and only if P is not a set of all edges of one dimension of Q d .…”