On super edge‐connectivity of Cartesian product graphs

Abstract

The super edge-connectivity λ of a connected graph G is the minimum cardinality of an edge-cut F in G such that every component of G − F contains at least two vertices. Let G i be a connected graph with order n i , minimum degree δ i and edge-connectivity λ i for i = 1, 2. This article shows that λ (G 1 × G 2 ) ≥ min{n 1 λ 2 , n 2 λ 1 , λ 1 + 2λ 2 , 2λ 1 +λ 2 } for n 1 , n 2 ≥ 3 and λ (K 2 ×G 2 ) = min{n 2 , 2λ 2 }, which generalizes the main result of Shieh on the super edge-connectedness of the Cartesian product of two regular graphs with maximum edge-connectivity. In particular, this article determines λ (G 1 × G 2 ) = min{n 1 δ 2 , n 2 δ 1 , ξ(G 1 × G 2 )} if λ (G i ) = ξ(G i ), where ξ(G) is the minimum edge-degree of a graph G.

Super restricted edge connected Cartesian product graphs

“…Using this results, certain classes of networks which are recursively defined by the Cartesian product can be simply shown to be super-λ , such as, n-dimensional toroidal mesh and n-dimensional Hypercube. The results in this paper improve results in [5]. The way that we prove the main results is similar to the way in [5].…”

“…The results in this paper improve results in [5]. The way that we prove the main results is similar to the way in [5].…”

“…If G is λ -optimal graph, then λ(G) = δ(G). [5].) λ (G 1 × G 2 ) = min{n 1 δ 2 , n 2 δ 1 , ξ(G 1 × G 2 )} if G 1 and G 2 are both λ -optimal.…”

“…In [1,9,[12][13][14], the authors considered the connectivity, super-connectivity and super-edgeconnectivity of Cartesian product graphs, and in [4,12,13], the authors introduced some results about connectivity and super (arc) connectivity of Cartesian product of digraphs. In [5,6,10], the authors studied the restricted (edge) connectivity of Cartesian product graphs and λ -optimal Cartesian product graphs. This article gives a sufficient condition for Cartesian product graphs to be super-λ .…”

Super-connected and super-arc-connected Cartesian product of digraphs

“…In 1957, Sabidussi [4] first discussed the connectivity of the Cartesian product of two undirected graphs. In [2,3,[5][6][7][8], the authors considered the connectivity, super-connectivity and super-edge-connectivity of Cartesian product graphs. To name a few, Chiue and Shieh [2] studied the super-connected and super-edgeconnected Cartesian product of two regular graphs, and in [6,7,9], the authors introduced some results about connectivity of Cartesian product of digraphs.…”