On the Edge Connectivity of Direct Products with Dense Graphs

Wang, Wei; Yan, Zhidan

doi:10.1007/s00373-012-1193-8

Cited by 6 publications

(7 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Mamut and Vumar [14] proved that κ(K m × K n ) = (m − 1)(n − 1) where m ≥ n ≥ 2. In [9], it was shown that if n ≥ 3 and G is a bipartite graph, then κ(G×K n ) = min{nκ(G), (n − 1)δ(G)}, and furthermore, the authors also conjectured that this is true for all nontrivial graph G. Later, this conjecture was confirmed independently by Wang and Wu [17] and Wang and Xue [18].…”

Section: Introductionmentioning

confidence: 92%

Super connectivity of direct product of graphs

Zhou

2015

Ars Math. Contemp.

View full text Add to dashboard Cite

For a graph G, κ(G) denotes its connectivity. A graph G is super connected, or simply super-κ, if every minimum separating set is the neighborhood of a vertex of G, that is, every minimum separating set isolates a vertex. The direct product G 1 ×G 2 of two graphs G 1 and G 2 is a graph with vertex set V (G 1 × G 2) = V (G 1) × V (G 2) and edge set E(G 1 × G 2) = {(u 1 , v 1)(u 2 , v 2) | u 1 u 2 ∈ E(G 1), v 1 v 2 ∈ E(G 2)}. Let Γ = G × K n , where G is a non-trivial graph and K n (n ≥ 3) is a complete graph on n vertices. In this paper, we show that Γ is not super-κ if and only if either κ(Γ) = nκ(G), or Γ ∼ = K , × K 3 (> 0).

show abstract

Section: Introductionmentioning

confidence: 92%

Super connectivity of direct product of graphs

Zhou

2015

Ars Math. Contemp.

View full text Add to dashboard Cite

show abstract

“…In [20] proved that G * is connected if G is connected and |S| < (n − 1)δ(G). In fact, when |S| = (n − 1)δ(G) here, we still get the result by using the same method in [20]. Proof.…”

Section: Lemma 1 ([21]mentioning

confidence: 99%

“…Guji and Vumar [10] presented the connectivity of Kronecker product of a bipartite graph and a complete graph and they proposed to investigate the connectivity of Kronecker product of a nontrivial graph and a complete graph. Recently, Wang and Xue [20] settled the problem and they obtained the following result.…”

Section: Introduction and Terminologymentioning

confidence: 99%

On the super connectivity of Kronecker products of graphs

Wang¹,

Shan²,

Wang³

2012

Information Processing Letters

Self Cite

View full text Add to dashboard Cite

Let G 1 and G 2 be two graphs. The Kronecker productA graph G is super connected, or simply super-κ, if every minimum separating set is the neighbors of a vertex of G, that is, every separating set isolates a vertex. In this paper we show that for an arbitrary graph G with κ(G) = δ(G) and K n (n ≥ 3) a complete graph on n vertices, G × K n is super-κ, where κ(G) and δ(G) are the connectivity and the minimum degree of G, respectively.

show abstract

“…Therefore it is not possible to express the connectivity of the direct product of graphs exclusively in terms of connectivities (and minimum degree, size and order) of the factors. There is an extensive list of articles on connectivity of direct products of graphs where the authors consider special examples and determine their connectivity, or they obtain upper and lower bounds for the connectivity, or they study related concepts such as super connectivity, see [2,3,5,6,10,11,14,15,16,7]. In this article we settle the question of edge connectivity of direct products as well as the question about the structure of a minimum edge-cut in the direct product of graphs.…”

Section: Introductionmentioning

confidence: 99%