Connectivity and edge-bipancyclicity of hamming shell

An Any graph obtained by deleting a Hamming code of length n from a n-cube Qn is called as a Hamming shell. It is well known that a Hamming shell is vertex-transitive, edge-transitive, distance preserving. Moreover, it is Hamiltonian and connected. In this paper, we prove that a Hamming shell is edge-bipancyclic and (n-1)-connected.Comment: Total 11 pages with 5 figure

Hamiltonian cycles in hypercubes with faulty edges

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 subgraph $T$ is disconnected halfway if at least half of its nodes have parity 0 (or 1, resp.) and the edges joining all nodes of parity 0 (or 1, resp.) in $T$ with nodes outside $T$, are faulty. The simplest examples of such traps are: (1) a vertex with $n-1$ incident faulty edges, or (2) a cycle $(u,v,w,x)$, where all edges going out of the cycle from $u$ and $w$ are faulty. In this paper we describe all traps disconnected halfway $T$ with the size $|T|\le8$, and discuss the problem whether there exist small sets of faulty edges which preclude Hamiltonian cycles and are not based on sets disconnected halfway. We describe heuristic which detects sets of faulty edges which preclude HC also those sets that are not based on subgraphs disconnected halfway. We describe all $Q_4$ cubes that are not Hamiltonian, and all $Q_5$ cubes with 8 or 9 faulty edges that are not Hamiltonian

Cycles in enhanced hypercubes

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 from $4$ to $2^n$; if $k$ is even, every edge of $Q_{n,k}$ also lies on a cycle of every odd length from $k+3$ to $2^n-1$, and some special edges lie on a shortest odd cycle of length $k+1$.Comment: 9 pages, 2 figure

The restricted hh-connectivity of balanced hypercubes

The restricted $h$-connectivity of a graph $G$, denoted by $\kappa^h(G)$, is defined as the minimum cardinality of a set of vertices $F$ in $G$, if exists, whose removal disconnects $G$ and the minimum degree of each component of $G-F$ is at least $h$. In this paper, we study the restricted $h$-connectivity of the balanced hypercube $BH_n$ and determine that $\kappa^1(BH_n)=\kappa^2(BH_n)=4n-4$ for $n\geq2$. We also obtain a sharp upper bound of $\kappa^3(BH_n)$ and $\kappa^4(BH_n)$ of $n$-dimension balanced hypercube for $n\geq3$ ($n\neq4$). In particular, we show that $\kappa^3(BH_3)=\kappa^4(BH_3)=12$

Structure and substructure connectivity of balanced hypercubes

The connectivity of a network directly signifies its reliability and fault-tolerance. Structure and substructure connectivity are two novel generalizations of the connectivity. Let $H$ be a subgraph of a connected graph $G$. The structure connectivity (resp. substructure connectivity) of $G$, denoted by $\kappa(G;H)$ (resp. $\kappa^s(G;H)$), is defined to be the minimum cardinality of a set $F$ of connected subgraphs in $G$, if exists, whose removal disconnects $G$ and each element of $F$ is isomorphic to $H$ (resp. a subgraph of $H$). In this paper, we shall establish both $\kappa(BH_n;H)$ and $\kappa^s(BH_n;H)$ of the balanced hypercube $BH_n$ for $H\in\{K_1,K_{1,1},K_{1,2},K_{1,3},C_4\}$.Comment: arXiv admin note: text overlap with arXiv:1805.0846

Fault-tolerance of balanced hypercubes with faulty vertices and faulty edges

Let $F_{v}$ (resp. $F_e$) be the set of faulty vertices (resp. faulty edges) in the $n$-dimensional balanced hypercube $BH_n$. Fault-tolerant Hamiltonian laceability in $BH_n$ with at most $2n-2$ faulty edges is obtained in [Inform. Sci. 300 (2015) 20--27]. The existence of edge-Hamiltonian cycles in $BH_n-F_e$ for $|F_e|\leq 2n-2$ are gotten in [Appl. Math. Comput. 244 (2014) 447--456]. Up to now, almost all results about fault-tolerance in $BH_n$ with only faulty vertices or only faulty edges. In this paper, we consider fault-tolerant cycle embedding of $BH_n$ with both faulty vertices and faulty edges, and prove that there exists a fault-free cycle of length $2^{2n}-2|F_v|$ in $BH_n$ with $|F_v|+|F_e|\leq 2n-2$ and $|F_v|\leq n-1$ for $n\geq 2$. Since $BH_n$ is a bipartite graph with two partite sets of equal size, the cycle of a length $2^{2n}-2|F_v|$ is the longest in the worst-case.Comment: 17 pages, 5 figures, 1 tabl

Paired many-to-many 2-disjoint path cover of balanced hypercubes with faulty edges

As a variant of the well-known hypercube, the balanced hypercube $BH_n$ was proposed as a novel interconnection network topology for parallel computing. It is known that $BH_n$ is bipartite. Assume that $S=\{s_1,s_2\}$ and $T=\{t_1,t_2\}$ are any two sets of two vertices in different partite sets of $BH_n$ ($n\geq1$). It has been proved that there exist two vertex-disjoint $s_1,t_1$-path and $s_2,t_2$-path of $BH_n$ covering all vertices of it. In this paper, we prove that there always exist two vertex-disjoint $s_1,t_1$-path and $s_2,t_2$-path covering all vertices of $BH_n$ with at most $2n-3$ faulty edges. The upper bound $2n-3$ of edge faults tolerated is optimal.Comment: 30 pages, 9 figure

Unpaired many-to-many disjoint path cover of balanced hypercubes

The balanced hypercube $BH_n$, a variant of the hypercube, was proposed as a desired interconnection network topology. It is known that $BH_n$ is bipartite. Assume that $S=\{s_1,s_2,\cdots,s_{2n-2}\}$ and $T=\{t_1,t_2,\cdots,t_{2n-2}\}$ are any two sets of vertices in different partite sets of $BH_n$ ($n\geq2$). It has been proved that there exists paired 2-disjoint path cover of $BH_n$. In this paper, we prove that there exists unpaired $(2n-2)$-disjoint path cover of $BH_n$ ($n\geq2$) from $S$ to $T$, which improved some known results. The upper bound $2n-2$ of the number of disjoint paths in unpaired $(2n-2)$-disjoint path cover is best possible.Comment: arXiv admin note: text overlap with arXiv:1804.0194

Minimum k-critical bipartite graphs

We study the problem of Minimum $k$-Critical Bipartite Graph of order $(n,m)$ - M$k$CBG-$(n,m)$: to find a bipartite $G=(U,V;E)$, with $|U|=n$, $|V|=m$, and $n>m>1$, which is $k$-critical bipartite, and the tuple $(|E|, \Delta_U, \Delta_V)$, where $\Delta_U$ and $\Delta_V$ denote the maximum degree in $U$ and $V$, respectively, is lexicographically minimum over all such graphs. $G$ is $k$-critical bipartite if deleting at most $k=n-m$ vertices from $U$ creates $G'$ that has a complete matching, i.e., a matching of size $m$. We show that, if $m(n-m+1)/n$ is an integer, then a solution of the M$k$CBG-$(n,m)$ problem can be found among $(a,b)$-regular bipartite graphs of order $(n,m)$, with $a=m(n-m+1)/n$, and $b=n-m+1$. If $a=m-1$, then all $(a,b)$-regular bipartite graphs of order $(n,m)$ are $k$-critical bipartite. For $a<m-1$, it is not the case. We characterize the values of $n$, $m$, $a$, and $b$ that admit an $(a,b)$-regular bipartite graph of order $(n,m)$, with $b=n-m+1$, and give a simple construction that creates such a $k$-critical bipartite graph whenever possible. Our techniques are based on Hall's marriage theorem, elementary number theory, linear Diophantine equations, properties of integer functions and congruences, and equations involving them

Matching preclusion and strong matching preclusion of the bubble-sort star graphs

Since a plurality of processors in a distributed computer system working in parallel, to ensure the fault tolerance and stability of the network is an important issue in distributed systems. As the topology of the distributed network can be modeled as a graph, the (strong) matching preclusion in graph theory can be used as a robustness measure for missing edges in parallel and distributed networks, which is defined as the minimum number of (vertices and) edges whose deletion results in the remaining network that has neither a perfect matching nor an almost-perfect matching. The bubble-sort star graph is one of the validly discussed interconnection networks related to the distributed systems. In this paper, we show that the strong matching preclusion number of an $n$-dimensional bubble-sort star graph $BS_n$ is $2$ for $n\geq3$ and each optimal strong matching preclusion set of $BS_n$ is a set of two vertices from the same bipartition set. Moreover, we show that the matching preclusion number of $BS_n$ is $2n-3$ for $n\geq3$ and that every optimal matching preclusion set of $BS_n$ is trivial.Comment: 12 pages, 5 figure