An implicit degree Dirac condition for Hamiltonian cycles. (Q2864472)

In this paper the authors deal with a concept of an implicit degree of a vertex in a graph \(G\) defined by \textit{Y. Zhu} et al. in [Graphs Comb. 5, No. 3, 283--290 (1989; Zbl 0701.05030)]. The implicit degree of a vertex generalizes the classical degree of a vertex and it has the classical degree as a lower bound. Zhu et al. [loc. cit.] proved sufficient conditions for Hamiltonicity of \(G\) in terms of the implicit degree of every vertex of \(G\), namely that every \(2\)-connected graph \(G\) of order \(n\) with \(\mathrm{id}(v)\geq \frac n2\) for each \(v\in V(G)\) contains a Hamiltonian cycle. In this paper, the authors show that if \(G\) is a \(2\)-connected graph of order \(n\) with the independence number \(\alpha (G)\leq \frac n2\) and \(\mathrm{id}(v)\geq \frac {n-1}2\) for every \(v\in V(G)\), then \(G\) is Hamiltonian with some exceptions.