Edge-fault-tolerant panconnectivity and edge-pancyclicity of the complete graph

doi:10.1016/j.ins.2013.02.012

Information Sciences

Volume 235, 20 June 2013, Pages 341-346

https://doi.org/10.1016/j.ins.2013.02.012 Get rights and content

Abstract

The complete graphs are an important class of graphs, and are also fundamental interconnection networks. Recently, Fu investigated their edge-fault-tolerant Hamiltonicity and Ho et al. investigated their edge-fault-tolerant Hamiltonian-connectivity. In this paper, we improve the result of Fu and point out that the proof of the result of Ho et al. fails. Then we consider the edge-fault-tolerant panconnectivity of the complete graphs and obtain the following result. Let F be any set of at most 2n − 10 faulty edges in the complete graph $K_{n}$ with n vertices, such that every vertex of the graph $G = K_{n} - F$ is incident with at least three edges and $G \notin {K_{8} - E (K_{4}), K_{10} - E (K_{5})}$ . Then G is nearly panconnected, i.e., for any two vertices u and v, there exists a path connecting u and v in G of any length from 3 to n − 1. As a corollary, every edge in the graph G lies on a cycle of any length from 4 to n. Moreover, the number 2n − 10 of faulty edges tolerated is sharp.

Introduction

One of the central issues in designing and evaluating an interconnection network is to study how well other existing networks can be embedded into this network. Linear arrays (i.e., paths) and rings (i.e., cycles), which are two fundamental networks for parallel and distributed computation, are suitable for developing simple algorithms with low communication cost. In this regard, embedding of paths and/or cycles of various lengths in networks is important and has become the area of intense study.

A graph is called Hamiltonian if it has a cycle containing all its vertices. A graph is called Hamiltonian-connected if there exists a Hamiltonian path connecting any two of its vertices. It is clear that any vertex in a Hamiltonian-connected graph on at least four vertices is incident with at least three edges. A graph G is called panconnected if for any two vertices u and v, there exists a path connecting u and v of any length l with $d_{G} (u, v) ⩽ l ⩽ | V (G) | - 1$ , where $d_{G} (u, v)$ denotes the distance between u and v in G, and $| V (G) |$ denotes the number of vertices in G. It is easy to see that Hamiltonian connectivity implies Hamiltonicity, and panconnectivity implies Hamiltonian connectivity. A graph G is called bipanconnected if for any two vertices u and v, there exists a path connecting u and v of any length l such that $d_{G} (u, v) ⩽ l ⩽ | V (G) | - 1$ and $l - d_{G} (u, v)$ is even. A (bipartite) graph G is said to have (bi) pancyclicity if it contains a cycle of any (even) length l with $g (G) ⩽ l ⩽ | V (G) |$ , where g(G) denotes the length of a shortest cycle in G. A (bipartite) graph is said to have edge-(bi) pancyclicity if any edge lies on a cycle of any (even) length l with $g (G) ⩽ l ⩽ | V (G) |$ . It is easy to see that edge-(bi) pancyclicity implies (bi) pancyclicity, and (bi) panconnectivity implies edge-(bi) pancyclicity.

Element (edge and/or vertex) failure is inevitable when a large parallel computer system is put in use. In this regard, the fault-tolerant capacity of a network is a critical issue in parallel computing. There have been studies of various networks with faulty elements, particularly on their path and/or cycle embedding properties. For (fault-tolerant) (bi) panconnectivity and (bi) pancyclicity of networks, see recent references [2], [5], [6], [8], [12], [15], [18], [19], [21], [22], [23], [25], [26], [28] and a survey [27]. For path and/or cycle embedding under other conditions in networks, see recent references [3], [4], [7], [9], [10], [13], [16], [17], [20], [24].

The complete graphs are an important class of graphs, and are also fundamental interconnection networks. Although much work has been done on the graph-theoretical properties of the complete graphs, the work on their fault-tolerance is relatively rare. Recently, Fu [11] investigated the edge-fault-tolerant Hamiltonicity of the complete graphs and Ho et al. [14] investigated their edge-fault-tolerant Hamiltonian connectivity. In this paper, we improve the result of Fu (see Corollary 1 below) and point out that the proof of the result of Ho et al. fails. Then we consider the edge-fault-tolerant panconnectivity on the complete graphs and obtain the following result.

Theorem 1

Let $K_{n}$ be the complete graph on $n (⩾ 6)$ vertices, and let F be any set of at most 2n − 10 faulty edges, such that every vertex of the graph $G = K_{n} - F$ is incident with at least three edges and $G \notin {K_{8} - E (K_{4}), K_{10} - E (K_{5})}$ . Then G is nearly panconnected, i.e., for any two vertices u and v, there exists a path connecting u and v in G of any length from 3 to n − 1. As a corollary, every edge in the graph G lies on a cycle of any length from 4 to n. Moreover, the number 2n − 10 of faulty edges tolerated is sharp.

Section snippets

Preliminaries

For graph-theoretical terminology and notation, we follow the ones given by Bondy and Murty [1]. Let $G = (V, E)$ denote a simple graph, where $V = V (G)$ is its vertex-set and $E = E (G)$ is its edge-set. The degree of a vertex v in a graph G is the number of edges incident with this vertex, which is denoted by $d_{G} (v)$ or simply $d (v)$ . The minimum degree of a graph G, denoted by $δ (G)$ , is the minimum value of the degrees of all its vertices. If $V (G) = {v_{1}, v_{2}, \dots, v_{n}}$ , then $d = (d (v_{1}), d (v_{2}), \dots, d (v_{n}))$ is called the degree

Some lemmas

In this section, we give some lemmas that are needed to show Theorem 2.

Lemma 1

see [11]

Let $K_{n}$ be the complete graph on $n (⩾ 5)$ vertices, and let $F \subset E (K_{n})$ with $δ (K_{n} - F) ⩾ 2$ , such that $| F | ⩽ 2 n - 8$ if $n \notin {7, 9}$ and $| F | ⩽ 2 n - 9$ if $n \in {7, 9}$ . Then the graph $K_{n} - F$ is Hamiltonian. Moreover, the number 2n − 8 (respectively, 2n − 9) of faulty edges tolerated is sharp if $n \notin {7, 9}$ (respectively, $n \in {7, 9}$ ).

Lemma 2

Let G be a graph on $n (⩾ 4)$ vertices and $w \in V (G)$ .

(1) Assume $d (w) ⩾ ⌈ n / 2 ⌉$ . If the graph $G^{'} = G - {w}$ is Hamiltonian, so is the graph G.
(2) Assume $d (w) ⩾$

Proof of Theorem 2 and a corollary

In this section, we give the proof of Theorem 2 and present a corollary.

Proof of Theorem 2

By Lemma 4, Lemma 9, to show Theorem 2 we only need to show that every edge in the graph $G = K_{n} - F$ lies on a cycle of any length from 5 to n − 1. Without loss of generality, we assume $| F | = 2 n - 10$ .

The following four claims can be easily proved.

Claim 1

Assume $δ (G) = 3$ . Then there exists exact one vertex $w \in V (G)$ with $d (w) = 3$ . Hence, $δ (G - {w}) ⩾ 3$ and the graph $K_{n} - {w}$ contains $n - 6 = (2 n - 10) - (n - 4) ⩽ 2 (n - 1) - 11$ faulty edges if $n ⩾ 7$ .

Claim 2

Assume $δ$

Conclusion remarks

In this paper, we investigate the edge-fault-tolerant Hamiltonicity and the edge-fault-tolerant panconnectivity of the complete graphs. The following two results are obtained: (1) If $F \subset E (K_{n})$ with $| F | ⩽ 2 n - 8$ and $δ (K_{n} - F) ⩾ 2$ , then the graph $K_{n} - F$ is Hamiltonian except the two graphs $K_{7} - E (K_{4})$ and $K_{9} - E (K_{5})$ . Moreover, the number 2n − 8 of faulty edges tolerated is sharp. (2) If $F \subset E (K_{n})$ with $| F | ⩽ 2 n - 10$ and $δ (K_{n} - F) ⩾ 3$ , then the graph $K_{n} - F$ is nearly panconnected except the two graphs $K_{8} - E (K_{4})$ and $K_{10} - E (K_{5})$ .

Acknowledgement

The author would like to thank the anonymous referees for their review comments that help to improve the original manuscript.

References (28)

X.-B. Chen
Some results on topological properties of folded hypercubes
Inform. Process. Lett.
(2009)
X.-B. Chen
Many-to-many disjoint paths in faulty hypercubes
Inform. Sci.
(2009)
X.-B. Chen
Cycles passing through a prescribed path in a hypercube with faulty edges
Inform. Process. Lett.
(2010)
X.-B. Chen
Edge-fault-tolerant diameter and bipanconnectivity of hypercubes
Inform. Process. Lett.
(2010)
Q. Dong et al.
Embedding paths and cycles in 3-ary n-cubes with faulty nodes and links
Inform. Sci.
(2010)
T. Dvořák et al.
Long paths in hypercubes with a quadratic number of faults
Inform. Sci.
(2009)
J. Fan et al.
Edge-pancyclicity and path-embeddability of bijective connection graphs
Inform. Sci.
(2008)
J.-F. Fang
The bipancycle-connectivity of the hypercube
Inform. Sci.
(2008)
J. Fink et al.
Long paths and cycles in hypercubes with faulty vertices
Inform. Sci.
(2009)
J.-S. Fu
Conditional fault Hamiltonicity of the complete graph
Inform. Process. Lett.
(2008)

P. Gregor et al.

Path partitions of hypercubes

Inform. Process. Lett.

(2008)

T.-Y. Ho et al.

Conditional fault hamiltonian connectivity of the complete graph

Inform. Process. Lett.

(2009)

S.-Y. Hsieh et al.

Edge-bipancyclicity of a hypercube with faulty vertices and edges

Disc. Appl. Math.

(2008)

T.-L. Kueng et al.

Long paths in hypercubes with conditional node-faults

Inform. Sci.

(2009)

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In this paper, the (bi)panconnectivity and edge-(bi)pancyclicity of $n$ -dimensional torus networks are investigated. An $n$ -dimensional torus $T = T (k_{1}, k_{2}, \dots, k_{n})$ with diameter $m_{0} = \sum_{i = 1}^{n} ⌊ k_{i} / 2 ⌋$ , where $k_{i} \geq 3$ for $i = 1, 2, \dots, n$ , is one of the most popular interconnection networks, the $k$ -ary $n$ -cube $Q_{n}^{k}$ ( $= T (k, k, \dots, k)$ ) is its special class. For any two vertices $u$ and $v$ in $T$ , we determine the set $ρ (u, v)$ of all lengths of $(u, v)$ -paths in $T$ by using path-shortening technique that can be used efficiently to construct the $(u, v)$ -paths in the torus $T$ . In particular, the following results are obtained: (1) The torus $T$ is bipanconnected and edge-bipancyclic; (2) If some $k_{j} \geq 3$ is odd and the other $k_{i} \geq 4$ is even for every $i \neq j$ , then the torus $T$ is $m_{1}$ -panconnected, where $m_{1} = (k_{j} - 1) / 2 + m_{0}$ , and the bound $m_{1}$ is optimal; (3) If both $k_{i} \geq 3$ and $k_{j} \geq 3$ are odd, then the torus $T$ is $m_{0}$ -panconnected, and the bound $m_{0}$ is optimal. (4) If some $k_{j}$ is odd, let $k$ be the minimum over all odd $k_{i}$ , then $T$ is $(k + 1)$ -edge pancyclic and the bound $k + 1$ is optimal if $T \neq Q_{n}^{k}$ , and $T$ is $h$ -edge-pancyclic and the bound $h = max {k - 1, 3}$ is optimal otherwise. Our results strengthen and generalize existing results.
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^☆: The work was supported by NSF of Fujian Province in China (Nos. 2010J01354 and 2011J01025).

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Edge-fault-tolerant panconnectivity and edge-pancyclicity of the complete graph☆

Abstract

Introduction

Section snippets

Preliminaries

Some lemmas

see [11]

Proof of Theorem 2 and a corollary

Conclusion remarks

Acknowledgement

Inform. Process. Lett.

Inform. Sci.

Inform. Process. Lett.

Inform. Process. Lett.

Inform. Sci.

Inform. Sci.

Inform. Sci.

Inform. Sci.

Inform. Sci.

Inform. Process. Lett.

Inform. Process. Lett.

Inform. Process. Lett.

Disc. Appl. Math.

Inform. Sci.