The H-force sets of the graphs satisfying the condition of Ore’s theorem

Abstract

Abstract Let G be a Hamiltonian graph. A nonempty vertex set X ⊆ V ( G ) X\subseteq V(G) is called a Hamiltonian cycle enforcing set (in short, an H-force set) of G if every X-cycle of G (i.e., a cycle of G containing all vertices of X) is a Hamiltonian cycle. For the graph G, h ( G ) h(G) (called the H-force number of G) is the smallest cardinality of an H-force set of G. Ore’s theorem states that an n-vertex graph G is Hamiltonian if d ( u ) + d ( v ) ≥ n d(u)+d(v)\ge n for every pair of nonadjacent vertices u , v u,v of G. In this article, we study the H-force sets of the graphs satisfying the condition of Ore’s theorem, show that the H-force number of these graphs is possibly n, or n − 2 n-2 , or n 2 \frac{n}{2} and give a classification of these graphs due to the H-force number.