A degree sum condition for long cycles passing through a linear forest

Let GG be a (k+m)(k+m)-connected graph and FF be a linear forest in GG such that |E(F)|=m|E(F)|=m and FF has at most k-2k-2 components of order 1, where k⩾2k⩾2 and m⩾0m⩾0. In this paper, we prove that if every independent set SS of GG with |S|=k+1|S|=k+1 contains two vertices whose degree sum is at least dd, then GG has a cycle CC of length at least min{d-m,|V(G)|}min{d-m,|V(G)|} which contains all the vertices and edges of FF.