The super edge-connectivity of direct product of a graph and a cycle

Abstract

The super edge-connectivity of a connected graph G, denoted by \({\lambda }'\left( G \right) \), if exists, is the minimum number of edges whose deletion disconnects the graph such that each component has no isolated vertices. The direct product of graphs G and H, denoted by \(G\times H\), is the graph with vertex set \(V\left( G\times H \right) =V\left( G \right) \times V\left( H \right) \), where two vertices \(\left( {{u}_{1}},{{v}_{1}} \right) \) and \(\left( {{u}_{2}},{{v}_{2}} \right) \) are adjacent in \(G\times H\) if and only if \({{u}_{1}}{{u}_{2}}\in E\left( G \right) \) and \({{v}_{1}}{{v}_{2}}\in E\left( H \right) \). In this paper, it is proved that \({\lambda }'\left( G\times {{C}_{n}} \right) = \min \{ 2n{\lambda }'\left( G \right) ,2\underset{xy\in E\left( G \right) }{{\min }}\,\left( {{\deg }_{G}}\left( x \right) +{{\deg }_{G}}\left( y \right) \right) -2 \}\) for (i) any connected graph G with \(\left| G \right| \le n\) or \(\Delta \left( G \right) \le n-1\) and an odd cycle \({{C}_{n}}\), or (ii) any split graph G with \(\left| G \right| \le n\) or \(\Delta \left( G \right) \le n-1\) and a cycle \({{C}_{n}}\).