Long Cycles Passing Through a Linear Forest

Abstract

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).