Long Cycles Passing Through a Linear Forest

A graph F is called a linear forest if $$V(F)=E(F)=\emptyset$$ or every component of F is a path. We denote by $$\omega _1(F)$$ the number of components of order 1 in F. In this article, we prove the following theorem. Let $$k\ge 5$$ and $$m\ge 0$$. Let G be a $$(k+m)$$-connected graph and F be a linear forest on a cycle of G with $$|E(F)|=m$$ and $$k + 1\le \omega _1(F) \le \lfloor \frac{4k-1}{3}\rfloor$$. Then G has a cycle of length at least $$\min \{\sigma _{2}(G)-m, |V(G)|\}$$ passing through F, where $$\sigma _{2}(G)$$ denotes the minimum degree sum of two independent vertices. Our result generalizes the theorem of Hu and Song (J Graph Theory 87(3):374–393).