A note on the connectivity of Kronecker products of graphs

Super connectivity of Kronecker products of graphs

“…Our proof follows the lines of the proof of Theorem 1.4 in [9], with modifications needed because of the super connectivity.…”

“…Mamut Kronecker product of two complete graphs [13]. And Guji and Vumar study the connectivity of the Kronecker product of a bipartite graph and a complete graph [9]. We also study the super connectivity of the Kronecker product of two complete graphs, complete graphs and paths, and complete graphs and cycles [8].…”

Rainbow vertex-connection and graph products

“…Here d e G (g, g ) represents the length of a shortest even walk between g and g in G, and d o G (g, g ) the length of a shortest odd walk between g and g in G. The formula was first shown in [23] and later in [16] in an equivalent version. There is no final solution for the connectivity of the direct product, only some partial results are known (see [5,18]). …”

On edge connectivity of direct products of graphs

“…Vertex connectivity of K m × K n has been determined in [13], in [8] Guji and Vumar established the vertex connectivity of G × K n for a bipartite graph G. Moreover in [20] the authors determined the edge connectivity of G × K 2 . In [4], the general case was considered, and the authors obtained an upper bound and a lower bound on the edge connectivity of the direct products of two graphs.…”

“…Let m, n be integers with n m 2 and n 3, then κ(K m × K n ) = (m − 1)(n − 1).Theorem 1.2. (See[8].) If G is a bipartite graph, then κ(G × K n ) = min{nκ(G), (n − 1)δ(G)} for n 3.0020-0190/$ -see front matter © 2011 Elsevier B.V. All rights reserved.…”