Proof of a conjecture of Bauer, Fan and Veldman

For a 1-tough graph $G$ we define $\sigma_3(G) = \min{\deg(u) + \deg(v)+ \deg(w):$ $\{u,v,w\}$ is an independent set of vertices$\}$ and $NC2(G)=\min {|N(u)\cup N(v)|: d(u,v)=2}$\mathrm{.}D\mathrm{.}Bauer,G\mathrm{.}FanandH\mathrm{.}J\mathrm{.}Veldmanprovedthat$c(G)\geq \min{n,2NC2(G)}$forany1-toughgraph$G$with$\sigma_3(G)\geq n\geq 3$,where$c(G)$isthecircumferenceof$G$ (D. Bauer, G. Fan and H.J.Veldman,Hamiltonian properties of graphs with large neighborhood unions,Discrete Mathematics, 1991). They also conjectured a stronger upper bound for the circumference: $c(G)\ge \min \{n,2NC2(G)+4\}$.In this paper, we prove this conjecture.