Long cycles containing k-ordered vertices in graphs

Let G be a (k+2)-connected graph on n vertices and S={v1,v2,…,vk} be any ordered set of vertices, that is, the vertices in S appear in the order of the sequence v1,v2,…,vk. We will show that if there exists a cycle containing S in the given order, then there exists a cycle C containing S in the given order such that |C|⩾min{n,σ2(G)} where σ2(G)=min{dG(u)+dG(v):u,v∈V(G);uv∉E(G)} when G is not complete, otherwise set σ2(G)=∞. This generalizes several related results known before.