Relative length of longest paths and longest cycles in triangle-free graphs

In this paper, we study triangle-free graphs. Let G=(V,E)G=(V,E) be an arbitrary triangle-free graph with minimum degree at least two and σ4(G)⩾|V(G)|+2σ4(G)⩾|V(G)|+2. We first show that either for any path PP in GG there exists a cycle CC such that |VP⧹VC|⩽1|VP⧹VC|⩽1, or GG is isomorphic to exactly one exception. Using this result, we show that for any set SS of at most δδ vertices in GG there is a cycle CC such that S⊆VCS⊆VC.