Partitioning series-parallel multigraphs into ν *-excluding edge covers

We prove that, for any given vertex ν* in a series-parallel graph G, its edge set can be partitioned into κ = min{κ′(G) + 1, δ(G)} subsets such that each subset covers all the vertices of G possibly except for ν*, where δ(G) is the minimum degree of G and κ′(G) is the edge-connectivity of G. In addition, we show that the results in this paper are best possible and a polynomial time algorithm can be obtained for actually finding such a partition by our proof.     

Cyclical games with prohibitions

We consider a certain combinatorial game on a digraph for two cases of the price function. For one case the game in question extends the cyclical game studied in Ehrenfeucht and Mycielski (1979) and Gurvitch, Karzanov and Khachiyan (1988) which, in its turn, is a generalization of the well-known problem of finding a minimum mean cycle in an edge-weighted digraph. We prove the existence of optimal uniform stationary strategies for both cases and give algorithms to find such strategies.This work was partially supported by Chaire municipale, Mairie de Grenoble.     

<Emphasis Type="Italic">k</Emphasis>-factors in regular graphs

Plesnik in 1972 proved that an （m - 1）-edge connected m-regular graph of even order has a 1-factor containing any given edge and has another 1-factor excluding any given m - 1 edges. Alder et al. in 1999 showed that if G is a regular （2n ＋ 1）-edge-connected bipartite graph, then G has a 1-factor containing any given edge and excluding any given matching of size n. In this paper we obtain some sufficient conditions related to the edge-connectivity for an n-regular graph to have a k-factor containing a set of edges and （or） excluding a set of edges, where 1 ≤ k ≤n/2. In particular, we generalize Plesnik＇s result and the results obtained by Liu et al. in 1998, and improve Katerinis＇ result obtained 1993. Furthermore, we show that the results in this paper are the best possible.     

Edge-splittings preserving local edge-connectivity of graphs

Let G=(V+s,E) be a 2-edge-connected graph with a designated vertex s. A pair of edges rs,st is called admissible if splitting off these edges (replacing rs and st by rt) preserves the local edge-connectivity (the maximum number of pairwise edge disjoint paths) between each pair of vertices in V. The operation splitting off is very useful in graph theory, it is especially powerful in the solution of edge-connectivity augmentation problems as it was shown by Frank [Augmenting graphs to meet edge-connectivity requirements, SIAM J. Discrete Math. 5(1) (1992) 22-53]. Mader [A reduction method for edge-connectivity in graphs, Ann. Discrete Math. 3 (1978) 145-164] proved that if d(s)≠3 then there exists an admissible pair incident to s. We generalize this result by showing that if d(s)?4 then there exists an edge incident to s that belongs to at least ⌊d(s)/3⌋ admissible pairs. An infinite family of graphs shows that this bound is best possible. We also refine a result of Frank [On a theorem of Mader, Discrete Math. 101 (1992) 49-57] by describing the structure of the graph if an edge incident to s belongs to no admissible pairs. This provides a new proof for Mader's theorem.     

On some structural properties of generalized fullerene graphs with 13 pentagonal faces

It is shown that every generalized fullerene graph G with 13 pentagons is 2-extendable, a brick, and cyclically 5-edge-connected, i.e., that G cannot be separated into two components, each containing a cycle, by deletion of fewer than five edges. New lower bound on the number of perfect matchings in such graphs are also established.     

Sufficient Conditions for Maximally Edge-connected and Super-edge-connected Digraphs Depending on the Size

Let D be a finite and simple digraph with vertex set V (D). The minimum degree δ of a digraph D is defined as the minimum value of its out-degrees and its in-degrees. If D is a digraph with minimum degree δ and edge-connectivity λ, then λ ≤ δ. A digraph is maximally edge-connected if λ=δ. A digraph is called super-edge-connected if every minimum edge-cut consists of edges incident to or from a vertex of minimum degree. In this note we show that a digraph is maximally edge-connected or super-edge-connected if the number of arcs is large enough.     

Optimization problems of the third edge-connectivity of graphs

The third edge-connectivity λ3(G) of a graph G is defined as the minimum cardinality over all sets of edges, if any, whose deletion disconnects G and each component of the resulting graph has at least 3 vertices. An upper bound has been established for λ3(G) whenever λ3(G) is well-defined. This paper first introduces two combinatorial optimization concepts, that is, maximality and superiority, of λ3(G), and then proves the Ore type sufficient conditions for G to be maximally and super third edge-connected. These concepts and results are useful in network reliability analysis.     

Partitioning series-parallel multigraphs intoν*-excluding edge covers

We prove that, for any given vertexν* in a series-parallel graph G, its edge set can be partitioned into k= min{k′(G) 1,δ(G)} subsets such that each subset covers all the vertices of G possibly except forν*, whereδ(G) is the minimum degree of G and k′(G) is the edge-connectivity of G. In addition, we show that the results in this paper are best possible and a polynomial time algorithm can be obtained for actually finding such a partition by our proof.     

On the minimum degree,edge-connectivity and connectivity of power graphs of finite groups

In this paper, the minimum degree of power graphs of certain cyclic groups, abelian p-groups, dihedral groups and dicyclic groups are obtained. It is ascertained that the edge-connectivity and minimum degree of power graphs are equal, and consequently, the minimum disconnecting sets of power graphs of the aforementioned groups are determined. In order to investigate the equality of connectivity and minimum degree of power graphs, certain necessary conditions for finite groups and a necessary and su?cient condition for finite cyclic groups are obtained. Moreover, the equality is discussed for the power graphs of abelian p-groups, dihedral groups and dicyclic groups.     

具有两个同阶轨道的双轨道图的圈边连通度

一个边割被称为圈边割,如果该边割能分离图的两个不同圈.如果一个图有圈边割,称该图为圈边可分离的.一个圈边可分离图G的最小圈边割的阶数被称为圈边连通度,记作cλ（G）.定义：ζ（G）=min{w（X）｜X导出G的最短圈},其中w（X）为端点分别在X和V（G）-X中的边的数目.如果一个圈边可分离图G使得cλ（G）=ζ（G）成立,称该图是圈边最优的.Tian和Meng在文章[11]以及Yang et al在文章[15]中研究了两种不同的双轨道图的圈边最优性.本文我们将研究具有两个同阶轨道的双轨道图的圈边连通度.