A degree sum condition for graphs to be prism hamiltonian

A degree sum condition for graphs to be covered by two cycles

Let G be a k-connected graph of order n. In [1], Bondy considered a degree sum condition for a graph to have a hamiltonian cycle, say, to be covered by one cycle. He proved that if σk+1(G) > (k+1)(n− 1)/2, then G has a hamiltonian cycle. On the other hand, concerning a degree sum condition for a graph to be covered by two cycles, Enomoto et al. [4] proved that if k = 1 and σ3(G) ≥ n, then G can...

Degree sum condition for Z3-connectivity in graphs

Let G be a 2-edge-connected simple graph on n vertices, let A denote an abelian group with the identity element 0, and let D be an orientation of G. The boundary of a function f : E(G) → A is the function ∂ f : V (G) → A given by ∂ f (v) = ∑ e∈E+(v) f (e) − ∑ e∈E−(v) f (e), where E(v) is the set of edges with tail v and E(v) is the set of edges with head v. A graph G is A-connected if for every...

A Degree Condition For Graphs to Have Connected (g, f)-Factors

A Fan-type condition for Hamiltonian graphs

For two non-adjacent vertices u: v in a graph G, we use a(u~ v) to denote the ma."\.imum cardinality of an independent vertex set of G containing both u and v. In this paper, we prove that if G is a 2-connected graph of order n and max{d{u),d(v}} ~ ~ for each pair of nonadjacent vertices Lt. v of G with 1 ::; IN( u) nN( v) I < a(u, v) -1, then either G is Hamiltonian or else G belongs to a fami...

A new sufficient condition for hamiltonian graphs

The study of Hamiltonian graphs began with Dirac’s classic result in 1952. This was followed by that of Ore in 1960. In 1984 Fan generalized both these results with the following result: If G is a 2-connected graph of order n and max{d(u), d(v)}≥n/2 for each pair of vertices u and v with distance d(u, v)=2, then G is Hamiltonian. In 1991 Faudree–Gould–Jacobson–Lesnick proved that if G is a 2-co...