Cubic \(1\)-Fault-Tolerant Hamiltonian Graphs, Globally \(3^{*}\)-Connected Graphs, and Super \(3\)-Spanning Connected Graphs

Abstract

A \(k\)-container \(C(u,v)\) in a graph \(G\) is a set of \(k\) internally
vertex-disjoint paths between vertices \(u\) and \(v\). A \(k^*\)-container
\(C(u,v)\) of \(G\) is a \(k\)-container such that \(C(u,v)\) contains all
vertices of \(G\). A graph is globally \(k^*\)-connected if there exists
a \(k^*\)-container \(C(u,v)\) between any two distinct vertices \(u\) and \(v\).
A \(k\)-regular graph \(G\) is super \(k\)-spanning connected if \(G\) is
\(i^*\)-connected for \(1 \leq i \leq k\). A graph \(G\) is \(1\)-fault-tolerant
Hamiltonian if \(G – F\) is Hamiltonian for any \(F \subseteq V(G)\) and
\(|F| = 1\). In this paper, we prove that for cubic graphs, every
super \(3\)-spanning connected graph is globally \(3^*\)-connected and
every globally \(3^*\)-connected graph is \(1\)-fault-tolerant Hamiltonian.
We present examples of super \(3\)-spanning connected graphs, globally
\(3^*\)-connected graphs that are not super \(3\)-spanning connected,
\(1\)-fault-tolerant Hamiltonian graphs that are globally \(1^*\)-connected
but not globally \(3^*\)-connected, and \(1\)-fault-tolerant Hamiltonian
graphs that are neither globally \(1^*\)-connected nor globally \(3^*\)-connected.
Furthermore, we prove that there are infinitely many graphs in each
such family.