Circumferences in 1-tough graphs (Q1903725)

For any finite, undirected graph \(G = (V(G), E(G))\) without loops or multiple edges, the circumference \(c(G)\) is the length of the longest cycle; \(G\) is 1-tough if, for every subset \(S \subseteq V(G)\) which disconnects \(G\), the number of components of \(G - S\) does not exceed \(|S |\). The author addresses a conjecture of \textit{D. Bauer}, \textit{H. J. Veldman}, \textit{A. Morgana} and \textit{E. F. Schmeichel} [Long cycles in graphs with large degree sums, Discrete Math. 79, No. 1, 59-70 (1990; Zbl 0713.05041)]: A 1-tough graph \(G\) on \(n > 3\) vertices, with minimum valency \(\delta \geq {n \over 3}\) has circumference \[ c(G) \geq \min \left\{ n, {3n + 1 \over 4} + {\delta \over 2} \right\} \geq {11n + 3 \over 12}. \] They observe that \textit{D. Bauer}, \textit{H. J. Veldman} and \textit{E. F. Schmeichel} [A generalization of a theorem of Bigalke and Jung, Ars. Comb. 26, 53-58 (1988; Zbl 0668.05040)] have shown that, under the given hypotheses, \[ c(G) \geq \min \left\{ n, {n \over 2} + \delta + 1 \right\} \geq {5n \over 6} + 1. \] Under the same hypotheses, the authors prove that \[ c(G) \geq \min \left\{ n, {2n + 1 + 2 \delta \over 3}, {3n + 2 \delta - 2 \over 4} \right\} \geq \min \left\{ {8n + 3 \over 9}, {11n - 6 \over 12} \right\}. \]