and

Let G be a triangle-free graph with δ(G) ≥ 2 and σ4(G) ≥ |V (G)|+ 2. Let S ⊂ V (G) consist of less than σ4/4 + 1 vertices. We prove the following. If all vertices of S have degree at least three, then there exists a cycle C containing S. Both the upper bound on |S | and the lower bound on σ4 are best possible