Hamiltonian Cycles Passing Through Prescribed Edges in Locally Twisted Cubes

2021 ◽  
Author(s):  
Dongqin Cheng
Keyword(s):  
Hamiltonian Cycle ◽  
Disjoint Paths ◽  
Hamiltonian Cycles ◽  
Induced Subgraph ◽  
Locally Twisted Cubes ◽  
Twisted Cube ◽  
Vertex Disjoint ◽  
Cube Formula ◽  
Twisted Cubes

Let [Formula: see text] be a set of edges whose induced subgraph consists of vertex-disjoint paths in an [Formula: see text]-dimensional locally twisted cube [Formula: see text]. In this paper, we prove that if [Formula: see text] contains at most [Formula: see text] edges, then [Formula: see text] contains a Hamiltonian cycle passing through every edge of [Formula: see text], where [Formula: see text]. [Formula: see text] has a Hamiltonian cycle passing through at most one prescribed edge.

scholarly journals Hamiltonian Cycles in Tough $(P_2\cup P_3)$-Free Graphs

10.37236/8657 ◽  
2021 ◽  
Vol 28 (1) ◽  
Author(s):  
Songling Shan
Keyword(s):  
Real Number ◽  
Disjoint Paths ◽  
Hamiltonian Cycles ◽  
Hard Problem ◽  
Free Graph ◽  
Induced Subgraph ◽  
Number Of Components ◽  
Np Hard Problem ◽  
Arbitrary Graphs ◽  
Vertex Disjoint

Let $t>0$ be a real number and $G$ be a graph. We say $G$ is $t$-tough if for every cutset $S$ of $G$, the ratio of $|S|$ to the number of components of $G-S$ is at least $t$. Determining toughness is an NP-hard problem for arbitrary graphs. The Toughness Conjecture of Chv\'atal, stating that there exists a constant $t_0$ such that every $t_0$-tough graph with at least three vertices is hamiltonian, is still open in general. A graph is called $(P_2\cup P_3)$-free if it does not contain any induced subgraph isomorphic to $P_2\cup P_3$, the union of two vertex-disjoint paths of order 2 and 3, respectively. In this paper, we show that every 15-tough $(P_2\cup P_3)$-free graph with at least three vertices is hamiltonian.

Three Types of Two-Disjoint-Cycle-Cover Pancyclicity and Their Applications to Cycle Embedding in Locally Twisted Cubes

2019 ◽  
Author(s):  
Tzu-Liang Kung ◽  
Hon-Chan Chen ◽  
Chia-Hui Lin ◽  
Lih-Hsing Hsu
Keyword(s):  
Disjoint Cycle ◽  
Cycle Cover ◽  
Disjoint Cycles ◽  
Locally Twisted Cubes ◽  
Twisted Cube ◽  
Vertex Disjoint ◽  
Twisted Cubes

Abstract A graph $G=(V,E)$ is two-disjoint-cycle-cover $[r_1,r_2]$-pancyclic if for any integer $l$ satisfying $r_1 \leq l \leq r_2$, there exist two vertex-disjoint cycles $C_1$ and $C_2$ in $G$ such that the lengths of $C_1$ and $C_2$ are $l$ and $|V(G)| - l$, respectively, where $|V(G)|$ denotes the total number of vertices in $G$. On the basis of this definition, we further propose Ore-type conditions for graphs to be two-disjoint-cycle-cover vertex/edge $[r_1,r_2]$-pancyclic. In addition, we study cycle embedding in the $n$-dimensional locally twisted cube $LTQ_n$ under the consideration of two-disjoint-cycle-cover vertex/edge pancyclicity.

scholarly journals DISTRIBUTED ALGORITHMS FOR BUILDING HAMILTONIAN CYCLES IN k-ARY n-CUBES AND HYPERCUBES WITH FAULTY LINKS

2007 ◽  
Vol 08 (03) ◽  
pp. 253-284 ◽  
Author(s):  
IAIN A. STEWART
Keyword(s):  
Distributed Algorithms ◽  
Distributed Algorithm ◽  
Hamiltonian Cycle ◽  
Sequential Algorithm ◽  
Hamiltonian Cycles ◽  
Cube Formula

We derive a sequential algorithm Find-Ham-Cycle with the following property. On input: k and n (specifying the k-ary n-cube [Formula: see text]); F, a set of at most 2n − 2 faulty links; and v , a node of [Formula: see text], the algorithm outputs nodes v + and v − such that if Find-Ham-Cycle is executed once for every node v of [Formula: see text] then the node v + (resp. v −) denotes the successor (resp. predecessor) node of v on a fixed Hamiltonian cycle in [Formula: see text] in which no link is in F. Moreover, the algorithm Find-Ham-Cycle runs in time polynomial in n and log k. We also obtain a similar algorithm for an n-dimensional hypercube with at most n − 2 faulty links. We use our algorithms to obtain distributed algorithms to embed Hamiltonian cycles k-ary n-cubes and hypercubes with faulty links; our hypercube algorithm improves on a recently-derived algorithm due to Leu and Kuo, and our k-ary n-cube algorithm is the first distributed algorithm for embedding a Hamiltonian cycle in a k-ary n-cube with faulty links.

scholarly journals A note on minimum linear arrangement for BC graphs

2018 ◽  
Vol 10 (02) ◽  
pp. 1850023
Author(s):  
Xiaofang Jiang ◽  
Qinghui Liu ◽  
N. Parthiban ◽  
R. Sundara Rajan
Keyword(s):  
Linear Ordering ◽  
Linear Arrangement ◽  
Arrangement Problem ◽  
Twisted Cube ◽  
Linear Arrangement Problem ◽  
Cube Formula ◽  
Twisted Cubes

A linear arrangement is a labeling or a numbering or a linear ordering of the vertices of a graph. In this paper, we solve the minimum linear arrangement problem for bijective connection graphs (for short BC graphs) which include hypercubes, Möbius cubes, crossed cubes, twisted cubes, locally twisted cube, spined cube, [Formula: see text]-cubes, etc., as the subfamilies.

scholarly journals Uniquely Hamiltonian Characterizations of Distance-Hereditary and Parity Graphs

10.37236/911 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Terry A. McKee
Keyword(s):  
Hamiltonian Cycle ◽  
Hamiltonian Cycles ◽  
Induced Subgraphs ◽  
Induced Subgraph ◽  
Odd Order ◽  
Close Relationship

A graph is shown to be distance-hereditary if and only if no induced subgraph of order five or more has a unique hamiltonian cycle; this is also equivalent to every induced subgraph of order five or more having an even number of hamiltonian cycles. Restricting the induced subgraphs to those of odd order five or more gives two similar characterizations of parity graphs. The close relationship between distance-hereditary and parity graphs is unsurprising, but their connection with hamiltonian cycles of induced subgraphs is unexpected.

The Tightly Super 2-extra Connectivity and 2-extra Diagnosability of Locally Twisted Cubes

2017 ◽  
Vol 17 (02) ◽  
pp. 1750006 ◽  
Author(s):  
YUNXIA REN ◽  
SHIYING WANG
Keyword(s):  
Fault Tolerance ◽  
Fault Diagnosis ◽  
Interconnection Network ◽  
Connected Subgraph ◽  
Minimum Cardinality ◽  
Pmc Model ◽  
Locally Twisted Cubes ◽  
Twisted Cube ◽  
Twisted Cubes

Connectivity plays an important role in measuring the fault tolerance of an interconnection network [Formula: see text]. A faulty set [Formula: see text] is called a g-extra faulty set if every component of G − F has more than g nodes. A g-extra cut of G is a g-extra faulty set F such that G − F is disconnected. The minimum cardinality of g-extra cuts is said to be the g-extra connectivity of G. G is super g-extra connected if every minimum g-extra cut F of G isolates one connected subgraph of order g + 1. If, in addition, G − F has two components, one of which is the connected subgraph of order g + 1, then G is tightly [Formula: see text] super g-extra connected. Diagnosability is an important metric for measuring the reliability of G. A new measure for fault diagnosis of G restrains that every fault-free component has at least (g + 1) fault-free nodes, which is called the g-extra diagnosability of G. The locally twisted cube LTQn is applied widely. In this paper, it is proved that LTQn is tightly (3n − 5) super 2-extra connected for [Formula: see text], and the 2-extra diagnosability of LTQn is 3n − 3 under the PMC model ([Formula: see text]) and MM* model ([Formula: see text]).

Embedding of Binomial Trees in Locally Twisted Cubes with Link Faults

2014 ◽  
Vol 1049-1050 ◽  
pp. 1736-1740
Author(s):  
Lan Tao You
Keyword(s):  
Parallel Computing ◽  
Interconnection Network ◽  
Binomial Tree ◽  
Binomial Trees ◽  
Locally Twisted Cubes ◽  
Twisted Cube ◽  
Twisted Cubes

As a newly introduced interconnection network for parallel computing, the locally twisted cube possesses many desirable properties. In this paper, binomial tree embeddings in locally twisted cubes are studied. We present two major results in this paper: (1) For any integern≥ 2, ann-dimensional binomial treeBncan be embedded inLTQnwith dilation 1 by randomly choosing any vertex inLTQnas the root. (2) For any integern≥ 2, ann-dimensional binomial treeBncan be embedded inLTQnwith up ton− 1 faulty links inlog(n− 1) steps where dilation = 1. The results are optimal in the sense that the dilations of all embeddings are 1.

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