On the Thomassen's conjecture (Q2746209)

Thomassen conjectured that a \(k\)-connected graph with stability number \(\alpha\) at least \(k\) has a cycle containing \(k\) independent vertices and all their neighbors. The author proves that if \(\alpha = k+3\) then any longest cycle has the stated property. This result follows from the fact that, for \(G\) with \(\alpha \geq k\), any longest cycle \(C\) for which \(G-C\) has no more than 3 components contains the desired vertices. Now if \(G\) has stability number \(k+3\), then it is known that for any longest cycle \(C\), \(G-C\) has no more than 3 components.