Long cycles through a linear forest

For a graph G and an integer k greater than or equal to1, let S(G)={x is an element ofV(G): d(G)(x)=0} and sigma (k)(G)=min{Sigma (k)(i=1) d(G)(v(i)) : {v(1),v(2),..., v(k)} is an independent set of G}. The main result of thi s paper is as follows. Let k greater than or equal to 3, m greater than or equal to 0, and 0 less than or equal to s less than or equal to 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 greater than or equal to min {\V(G)\, 2/k sigma (k)(G)-m} passing t hrough E(F) boolean OR V(F). This generalizes three related results known p reviously. (C) 2001 Academic Press.