Related papers: A perfect matching reciprocity method for embeddin…
The augmented cube $AQ_n$, proposed by Choudum and Sunitha [S. A. Choudum, V. Sunitha, Augmented cubes, Networks 40 (2) (2002) 71-84], is a $(2n-1)$-regular $(2n-1)$-connected graph $(n\ge 4)$. This paper determines that the 2-extra…
The spined cube $SQ_n$ is a variant of the hypercube $Q_n$, introduced by Zhou et al. in [Information Processing Letters 111 (2011) 561-567] as an interconnection network for parallel computing. A graph $\G$ is an $m$-Cayley graph if its…
In this paper, we consider the problem of decomposing the augmented cube $AQ_n$ into two spanning, regular, connected and pancyclic subgraphs. We prove that for $ n \geq 4$ and $ 2n - 1 = n_1 + n_2 $ with $ n_1, n_2 \geq 2,$ the augmented…
The $n$-dimensional hypercube graph $Q_n$ has as vertices all subsets of $\{1, \ldots, n\}$, and an edge between any two sets that differ in a single element. The Ruskey-Savage conjecture states that every matching of the $n$-dimensional…
We consider hypercubes with pairwise disjoint faulty edges. An $n$-dimensional hypercube $Q_n$ is an undirected graph with $2^n$ nodes, each labeled with a distinct binary strings of length $n$. The parity of the vertex is 0 if the number…
The hypercube is one of the most popular interconnection networks since it has simple structure and is easy to implement. The $n$-dimensional twisted cube, denoted by $TQ_n$, an important variation of the hypercube, possesses some…
The hypercube Qn of dimension n is one of the most versatile and powerful interconnection networks. The n-dimensional folded cube denoted as FQn, a variation of the hypercube possesses some embeddable properties that the hypercube does not…
Let $n\geq 2$ be an integer, and let $i\in\{0,...,n-1\}$. An $i$-th dimension edge in the $n$-dimensional hypercube $Q_n$ is an edge ${v_1}{v_2}$ such that $v_1,v_2$ differ just at their $i$-th entries. The parity of an $i$-th dimension…
The $n$-dimensional hypercube network $Q_n$ is one of the most popular interconnection networks since it has simple structure and is easy to implement. The $n$-dimensional locally twisted cube, denoted by $LTQ_n$, an important variation of…
Connectivity is a cornerstone concept in graph theory, essential for evaluating the robustness of networks against failures. To better capture fault tolerance in complex systems, researchers have extended classical connectivity notions, one…
The $d$-dimensional hypercube graph $Q_d$ has as vertices all subsets of $\{1,\ldots,d\}$, and an edge between any two sets that differ in a single element. The Ruskey-Savage conjecture asserts that every matching of $Q_d$, $d\ge 2$, can be…
Dong and Wang in [Theor. Comput. Sci. 771 (2019) 93--98] conjectured that the resulting graph of the $n$-dimensional folded hypercube $FQ_n$ by deleting any perfect matching is isomorphic to the hypercube $Q_n$. In this paper, we show that…
The hypercube \( Q_n \) contains a Hamiltonian path joining \( x \) and \( y \) (where $x$ and $y$ from the opposite partite set) containing \( P \) if and only if the induced subgraph of \( P \) is a linear forest, where none of these…
We study Hamiltonicity in random subgraphs of the hypercube $\mathcal{Q}^n$. Our first main theorem is an optimal hitting time result. Consider the random process which includes the edges of $\mathcal{Q}^n$ according to a uniformly chosen…
Szepietowski [A. Szepietowski, Hamiltonian cycles in hypercubes with $2n-4$ faulty edges, Information Sciences, 215 (2012) 75--82] observed that the hypercube $Q_n$ is not Hamiltonian if it contains a trap disconnected halfway. A proper…
The balanced hypercube, $BH_n$, is a variant of hypercube $Q_n$. R.X. Hao et al. $(2014)$ \cite{R.X.Hao} showed that there exists a fault-free Hamiltonian path between any two adjacent vertices in $BH_n$ with $(2n-2)$ faulty edges. D.Q.…
\noindent The augmented cube graph $AQ_n$ is the Cayley graph of $\mathbb{Z}_2^n$ with respect to the set of $2n-1$ generators $\{e_1,e_2, \ldots,e_n, 00\ldots0011, 00\ldots0111, 11\ldots1111 \}$. It is known that the order of the…
Faulty networks are useful because link or node faults can occur in a network. This paper examines the Hamiltonian properties of hypercubes under certain conditional faulty edges. Let consider the hypercube \( Q_n \), for \( n \geq 5 \) and…
Ruskey and Savage asked the following question: Does every matching of $Q_{n}$ for $n\geq2$ extend to a Hamiltonian cycle of $Q_{n}$? J. Fink showed that the question is true for every perfect matching, and solved the Kreweras' conjecture.…
The enhanced hypercube $Q_{n,k}$ is a variant of the hypercube $Q_n$. We investigate all the lengths of cycles that an edge of the enhanced hypercube lies on. It is proved that every edge of $Q_{n,k}$ lies on a cycle of every even length…