Stability, Total Vertices, and Hamiltonian Cycles

A known theorem of Bigalke and Jung says that the only nonhamiltonian, tough graph \(G\) with \(\alpha(G) \leq H(G) + 1\), where \(H(G) \geq 3\), is the Petersen graph. In this paper we characterize all nonhamiltonian, tough graphs having k total vertex (i.e. adjacent to all others) with \(\alpha(G) \leq k+ 2\) (Theorem 3).