Bauer–Fan–Veldman circumference conjecture for 1-tough graphs

Let $G$ be a graph of order $n$, let $\tau(G)$ denote its toughness, let $\rho(G)$ denote the independence number, and let $\sigma_3(G)$ be its smallest union degree of order $3$:

$$\sigma_3(G)=\begin{cases}\min\{\deg(v_1)+\deg(v_2)+\deg(v_3):\{v_1,v_2,v_3\}\text{ is independent}\},&\rho(G)\geq 3,\\3(n-1),&\rho(G)<3.\end{cases}$$

For nonadjacent vertices $u,v$ at distance $2$, write $N(u,v)=N(u)\cup N(v)$, and define

$$NC2(G)=\begin{cases}\min\{|N(u)\cup N(v)|:d(u,v)=2\},&G\text{ is not complete},\ -1,&G\text{ is complete}.\end{cases}$$

Let $c(G)$ be the circumference of $G$, namely the length of a longest cycle. Bauer–Fan–Veldman conjecture. If $G$ is a 1-tough graph with $\sigma_3(G)\geq n\geq 3$, then

$$c(G)\geq\min\{n,2NC2(G)+4\}.$$

This strengthens the earlier lower bound $c(G)\geq\min\{n,2NC2(G)\}$; the intermediate bound with additive constant $2$ follows from a theorem of Hoa. The conjecture is the claim proved in the source paper.