# Bound on m-restricted Edge Connectivity

Bound on m-restricted Edge Connectivity An m-restricted edge cut is an edge cut that separates a connected graph into a disconnected one with no components having order less than m. m-restricted edge connectivity λ m is the cardinality of a minimum m-restricted edge cut. Let G be a connected k-regular graph of order at least 2m that contains m-restricted edge cuts and X be a subgraph of G. Let ∂(X) denote the number of edges with one end in X and the other not in X and ξ m = min{∂(X) : X is a connected vertex-induced subgraph of order m}. It is proved in this paper that if G has girth at least m/2+ 2, then λ m ≤ ξ m . The upper bound of λ m is sharp. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematicae Applicatae Sinica Springer Journals

# Bound on m-restricted Edge Connectivity

, Volume 19 (3) – Mar 3, 2017

## Bound on m-restricted Edge Connectivity

Acta Mathematicae Applicatae Sinica, English Series Vol. 19, No. 3 (2003) 505–510 1,2 2 Jian-ping Ou ,Fu-ji Zhang Department of Mathematics, Zhangzhou Normal College, Fujian 363000, China (E-mail: oujp@263.net) Department of Mathematics, Xiamen University, Xiamen 361005, China (E-mail: fjzhang@jingxian.xmu.edu.cn) Abstract An m-restricted edge cut is an edge cut that separates a connected graph into a disconnected one with no components having order less than m. m-restricted edge connectivity λ is the cardinality of a minimum m-restricted edge cut. Let G be a connected k-regular graph of order at least 2m that contains m-restricted edge cuts and X be a subgraph of G.Let ∂(X) denote the number of edges with one end in X and the other not in X and ξ =min{∂(X): X is a connected vertex-induced subgraph of order m}.It is proved in this paper that if G has girth at least m/2+ 2, then λ ≤ ξ . The upper bound of λ is sharp. m m m Keywords Regular graph, bound, restricted edge connectivity 2000 MR Subject Classiﬁcation 05C40 1 Introduction Let G be a connected k-regular graph of order at least 2m,where m is an integer with m ≥ 2. Edge cut S of graph G is called an m-restricted edge cut if G − S contains no components of order less than m. m-restricted edge connectivity λ is the cardinality of minimum m-restricted edge cut. These concepts are generalization of so-called restricted edge cut and restricted edge [3] connectivity, which were proposed by Esfahanian and Hakimi to estimate more precisely the tolerance and reliability of multiple processors and communication networks. Let N and M be two multiple processors (or communication networks) with the same number of nodes. Under some reasonable conditions, it is proved that if every link has the same small enough failure probability and λ (N)= λ (M)for all i<p but λ (N ) <λ (M ), i i p p [5] then M is more reliable than N when...

/lp/springer_journal/bound-on-m-restricted-edge-connectivity-e6Q4L5JX0u
Publisher
Springer Journals
Copyright © 2003 by Springer-Verlag Berlin Heidelberg
Subject
Mathematics; Applications of Mathematics; Math Applications in Computer Science; Theoretical, Mathematical and Computational Physics
ISSN
0168-9673
eISSN
1618-3932
D.O.I.
10.1007/s10255-003-0127-x
Publisher site
See Article on Publisher Site

### Abstract

An m-restricted edge cut is an edge cut that separates a connected graph into a disconnected one with no components having order less than m. m-restricted edge connectivity λ m is the cardinality of a minimum m-restricted edge cut. Let G be a connected k-regular graph of order at least 2m that contains m-restricted edge cuts and X be a subgraph of G. Let ∂(X) denote the number of edges with one end in X and the other not in X and ξ m = min{∂(X) : X is a connected vertex-induced subgraph of order m}. It is proved in this paper that if G has girth at least m/2+ 2, then λ m ≤ ξ m . The upper bound of λ m is sharp.

### Journal

Acta Mathematicae Applicatae SinicaSpringer Journals

Published: Mar 3, 2017

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