Longest cycles in 4-connected graphs

Let ${\sigma }_4^1=min\text{{}{\sum }_{i=1}^{4}d\left(v_i\right)+\left|{?}_{i=1}^{4}N\left(v_i\right)\right|?\left|{?}_{i=1}^{4}N\left(v_i\right)\right|:\{v_1,v_2,v_3,v_4\}$ is an independent set of a graph G$\text{}}$. In this paper, we give a low bound for the length of a longest cycle in a 4-connected graph and get the following result: If $G$ is a 4-connected graph on $n$ vertices, then the circumference $c\left(G\right)\ge min\{n,{\sigma }_4^1∕2\}$. Moreover, we give graphs to show that the connectivity in our result is best possible with respect to the low bound and the low bound in our result is also best possible with respect to the connectivity.