Let D be a finite and simple digraph with vertex set V(D).The minimum degreeδof a digraph D is defined as the minimum value of its out-degrees and its in-degrees.If D is a digraph with minimum degreeδand edge-connec...Let D be a finite and simple digraph with vertex set V(D).The minimum degreeδof a digraph D is defined as the minimum value of its out-degrees and its in-degrees.If D is a digraph with minimum degreeδand edge-connectivity λ,then λ≤δ.A digraph is maximally edge-connected ifλ=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.展开更多
The edge-connectivity of a graph or a hypergraph is defined as the minimum number of edges whose removal renders the graph or hypergraph disconnected.A graph or hypergraph is called maximally edge-connected if the edg...The edge-connectivity of a graph or a hypergraph is defined as the minimum number of edges whose removal renders the graph or hypergraph disconnected.A graph or hypergraph is called maximally edge-connected if the edge-connectivity equals its minimum degree.In this paper,we show that some classical sufficient conditions for graphs to be maximally edge-connected can be generalized to hypergraphs.展开更多
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 ...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.展开更多
The h-super connectivity κh and the h-super edge-connectivity λh are more refined network reliability indices than the conneetivity and the edge-connectivity. This paper shows that for a connected balanced digraph D...The h-super connectivity κh and the h-super edge-connectivity λh are more refined network reliability indices than the conneetivity and the edge-connectivity. This paper shows that for a connected balanced digraph D and its line digraph L, if D is optimally super edge-connected, then κ1（L） = 2λ1 （D）, and that for a connected graph G and its line graph L, if one of κ1 （L） and λ（G） exists, then κ1（L） = λ2（G）. This paper determines that κ1（B（d, n） is equal to 4d- 8 for n = 2 and d ≥ 4, and to 4d-4 for n ≥ 3 and d ≥ 3, and that κ1（K（d, n）） is equal to 4d- 4 for d 〉 2 and n ≥ 2 except K（2, 2）. It then follows that B（d,n） and K（d, n） are both super connected for any d ≥ 2 and n ≥ 1.展开更多
文摘Let D be a finite and simple digraph with vertex set V(D).The minimum degreeδof a digraph D is defined as the minimum value of its out-degrees and its in-degrees.If D is a digraph with minimum degreeδand edge-connectivity λ,then λ≤δ.A digraph is maximally edge-connected ifλ=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.
基金This research was partially supported by the National Natural Science Foundation of China(Nos.11571222 and 11871329).
文摘The edge-connectivity of a graph or a hypergraph is defined as the minimum number of edges whose removal renders the graph or hypergraph disconnected.A graph or hypergraph is called maximally edge-connected if the edge-connectivity equals its minimum degree.In this paper,we show that some classical sufficient conditions for graphs to be maximally edge-connected can be generalized to hypergraphs.
基金This work was supported by the National Natural Science Foundation of China(Grant No.10471131)the Natural Science Foundation of Zhejiang Province(Grant No.102055).
文摘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.
基金Supported by the National Natural Science Foundation of China(No.10271114,No.10301031).
文摘The h-super connectivity κh and the h-super edge-connectivity λh are more refined network reliability indices than the conneetivity and the edge-connectivity. This paper shows that for a connected balanced digraph D and its line digraph L, if D is optimally super edge-connected, then κ1（L） = 2λ1 （D）, and that for a connected graph G and its line graph L, if one of κ1 （L） and λ（G） exists, then κ1（L） = λ2（G）. This paper determines that κ1（B（d, n） is equal to 4d- 8 for n = 2 and d ≥ 4, and to 4d-4 for n ≥ 3 and d ≥ 3, and that κ1（K（d, n）） is equal to 4d- 4 for d 〉 2 and n ≥ 2 except K（2, 2）. It then follows that B（d,n） and K（d, n） are both super connected for any d ≥ 2 and n ≥ 1.
