An Ore-type Condition for Cyclability

Let G = (V (G), E(G)) be a finite simple graph without loops. The neighbourhood N (v) of a vertex v is the set of vertices adjacent to v. The degree d(v) of v is |N (v)|. The minimum and maximum degree of G are denoted by δ(G) and 1(G), respectively. For a vertex v ∈ V (G) and a subset S ⊆ V (G), NS(v) is the set of neighbours of v contained in S, i.e., NS(v) = N (v) ∩ S. We let dS(v) = |NS(v)|. For S ⊆ V (G), G[S] denotes the subgraph induced by S. Let H be a subgraph of G. If h1h2 ∈ E(G) for any h1, h2 ∈ V (H), then we say H is a clique. A path with one end u is called a u-path. Let u, v ∈ V (G). A spanning subgraph H of G is called a (u, v)-path-factor if H contains two components, one of them is a u-path and the other is a v-path. Let P be a path. We denote by −→ P the path P with a given direction, and by ←− P the path P with the reverse direction. If u, v ∈ V (P), then u −→ P v denotes the consecutive vertices of P from u to v in the direction specified by −→ P . The same vertices, in reverse order, are given by v ←− P u. If a path or cycle includes every vertex of V (G), then it is called a Hamilton path or cycle. If G contains a Hamilton cycle, then we say G is Hamiltonian. Furthermore, we define