On 3-Extra Connectivity and 3-Extra Edge Connectivity of Folded Hypercubes

Abstract

Given a graph ${\mbi{G}}$ and a non-negative integer ${{g}}$ , the ${{g}}$ -extra connectivity (resp. ${{g}}$ -extra edge connectivity) of ${\mbi{G}}$ is the minimum cardinality of a set of vertices (resp. edges) in ${\mbi{G}}$ , if it exists, whose deletion disconnects ${\mbi{G}}$ and leaves each remaining component with more than ${{g}}$ vertices. This study shows that the 3-extra connectivity (resp. 3-extra edge connectivity) of an ${\mbi{n}}$ -dimensional folded hypercube is ${4}{{n}} - {5}$ for ${{n}} \geq {6}$ (resp. ${4}{{n}} - {4}$ for ${{n}} \geq {5}$ ). This study also provides an upper bound for the ${{g}}$ -extra connectivity on folded hypercubes for ${{g}} \geq {6}$ .