An existence criterion for a cycle such that the vertex set beyond this cycle is independent

Abstract

We prove that if $G$ is a 2-connected graph with $\delta (G)⩾\frac{v(G)+2}{3}$ then $G$ has a cycle $W$ such that $V(G-W)$ is independent. This result is best possible in the sense that it becomes false if $\frac{v(G)+2}{3}$ is replaced by any smaller number.