A neighborhood union condition for pancyclic graphs (Q2721833)

Let \(G\) be a \(2\)-connected graph of order \(n\geq 31\). It is shown that if \(|N(u)\cup N(v)|\geq (2n- 3)/3\) for each pair of nonadjacent vertices \(u\) and \(v\) of \(G\), then \(G\) is pancyclic (i.e. \(G\) contains cycles of each length from \(3\) to \(n\)). This improves results of \textit{D. Bauer}, \textit{G. Fan}, and \textit{H. J. Veldman} [Discrete Math. 96, No. 1, 33-49 (1991; Zbl 0741.05039)] and \textit{R. J. Faudree}, \textit{R. J. Gould}, \textit{M. S. Jacobson}, and \textit{L. Lesniak} [Ars Comb. 31, 139-148 (1991; Zbl 0739.05056)], and an example is described which implies the result is sharp.