Long Cycles through a Linear Forest

For a graph G and an integer k⩾1, let S(G)={x∈V(G) : dG(x)=0} and σk(G)=min{∑ki=1 dG(vi) : {v1, v2, …, vk} is an independent set of G}. The main result of this paper is as follows. Let k⩾3, m⩾0, and 0⩽s⩽k−3. Let G be a (m+k−1)-connected graph and let F be a subgraph of G with |E(F)|=m and |S(F)|=s. If every component of F is a path, then G has a cycle of length ⩾min{|V(G)|, 2kσk(G)−m} passing through E(F)∪V(F). This generalizes three related results known previously.