Long paths with endpoints in given vertex-subsets of graphs

Let G=(V,E)G=(V,E) be a connected graph of order n, t   a real number with t⩾1t⩾1 and M⊆V(G)M⊆V(G) with |M|⩾nt⩾2. In this paper, we study the problem of some long paths to maintain their one or two different endpoints in M  . We obtain the following two results: (1) for any vertex v∈V(G)v∈V(G), there exists a vertex u∈Mu∈M and a path P with the two endpoints v and u   to satisfy |V(P)|⩾min{44+tdG(u)+4-2t4+t, 21+tdG(u)-1, dG(u)+1-t}dG(u)+1-t}; (2) there exists either a cycle C to cover all vertices of M or a path P   with two different endpoints u0u0 and upup in M   to satisfy |V(P)|⩾min{n,f(t)1+f(t)(dG(u0)+dG(up))-2t-61+f(t)}, where f(t)=min{4t,2t-1}.