A Hamiltonian Condition Through A Path System

Let G be a graph with order n.If G contains a cycle passing through all its vertices,then G is called Hamiltonian .A path system is a graph each of whose components is a path.Define σ₃(G) as the minimum of d{x₁) + d(x₂) + d(x₃) — |N(x₁)∩N(x₂)∩N(x₃)|,where x₁, x₂, x₃ are any 3 pairwise nonadjacent vertices in G. In 1991 E.Flandrin et al. prove that for a 2-connected graph G with order n,if σ₃ (G) ≥ n,then G is hamiltonian.In 2004 E.Flandrin et al. give a regional version of the above theorem concerning passing through a fixed vertex set.They prove that for a 2k-connected graph G with order n, let X₁, X₂, ...,X_k be k subsets of V(G) with X=X₁∪X₂∪ ... ∪ X_k. If for each i, 1 ≤ i ≤ k, σ₃(X_i) ≥ n, then G has a hamiltonian cycle passing through X.In this paper, given a (m+2)-connected graph G and a path system M C G with |E{M)| = m,we show that if σ₃(G) ≥ n + rn,then G has a Hamilton cycle passing through M.And the condition σ₃(G) ≥ n + m is sharp.