On one extension of Dirac’s theorem on Hamiltonicity

Abstract

The classical Dirac theorem asserts that every graph G on n≥3 vertices with minimum degree δ(G)≥⌈n∕2⌉ is Hamiltonian. The lower bound of ⌈n∕2⌉ on the minimum degree of a graph is tight. In this paper, we extend the classical Dirac theorem to the case where δ(G)≥⌊n∕2⌋ by identifying the only non-Hamiltonian graph families in this case. We first present a short and simple proof. We then provide an alternative proof that is constructive and self-contained. Consequently, we provide a polynomial-time algorithm that constructs a Hamiltonian cycle, if exists, of a graph G with δ(G)≥⌊n∕2⌋, or determines that the graph is non-Hamiltonian. Finally, we present a self-contained proof for our algorithm which provides insight into the structure of Hamiltonian cycles when δ(G)≥⌊n∕2⌋ and is promising for extending the results of this paper to the cases with smaller degree bounds.