The numbers of 2-connected and 2-edge-connected Steinhaus graphs with n vertices are determined for n¿ 6. The possible generating strings of 2-connected and 2-edge-connected Steinhaus graphs are classi0ed. c © 2002 Elsevier Science B.V. All rights reserved.

In 1996, Ota and Tokuda showed that a star-free graph with sufficiently high minimum degree admits a 2-factor. More recently it was shown that the minimum degree condition can be significantly reduced if one also requires that the graph is not only star-free but also of sufficiently high edge-connectivity. In this paper we reduce the minimum degree condition further for star-free graphs that al...

For a connected graphG= (V ,E), an edge set S ⊂ E is a k-restricted-edge-cut, ifG−S is disconnected and every component of G− S has at least k vertices. The k-restricted-edge-connectivity of G, denoted by k(G), is defined as the cardinality of a minimum k-restricted-edge-cut. The k-isoperimetric-edge-connectivity is defined as k(G)=min{|[U,U ]| : U ⊂ V, |U | k, |U | k}, where [U,U ] is the set ...

In this paper, we study the relationship between eigenvalues and the existence of certain subgraphs in regular graphs. We give a condition on an appropriate eigenvalue that guarantees a lower bound for the matching number of a t-edge-connected d-regular graph, when t ≤ d − 2. This work extends some classical results of von Baebler and Berge and more recent work of Cioabă, Gregory, and Haemers. ...

A graph X is max-λ if λ(X) = δ(X). A graph X is super-λ if X is max-λ and every minimum edge-cut set of X isolates one vertex. In this paper, we proved that for all but a few exceptions, the mixed Cayley graph which is defined as a new kind of semi-regular graph is max-λ and super-λ. c © 2008 Elsevier B.V. All rights reserved.

The pyramid network is a hierarchy structure based on meshes. Chen et al. in 2004 proposed a new hierarchy structure, called the enhanced pyramid network, by replacing each mesh at layer greater than one of the pyramid network with a torus. This investigation demonstrates that the enhanced pyramid network with two faulty edges is Hamiltonian. The result is optimal because both edge connectivity...

In this paper we study the average distance in weighted graphs. More precisely, we consider assignments of families of non-negative weights to the edges. The aim is to maximise (minimise, respectively) the average distance in the resulting weighted graph. Two variants of the problem are considered depending on whether the collection of weights is fixed or not. The main results of this paper are...

Given two nonnegative integers s and t , a graph G is (s, t)-supereulerian if for any disjoint sets X, Y ⊂ E(G) with |X | ≤ s and |Y | ≤ t , there is a spanning eulerian subgraph H of G that contains X and avoids Y . We prove that if G is connected and locally k-edge-connected, thenG is (s, t)-supereulerian, for any pair of nonnegative integers s and t with s+t ≤ k−1. We further show that if s ...

Splitting off a pair of edges su; sv in a graph G means replacing these two edges by a new edge uv. This operation is well-known in graph theory. Let G = (V + s; E + F ) be a graph which is k-edge-connected in V and suppose that jF j is even. Here F denotes the set of edges incident with s. Lovász [12] proved that if k 2 then the edges in F can be split off in pairs preserving the k-edge-connec...