Long Cycles in 1-tough Graphs

In 1952, Dirac proved that every 2-connected graph with minimum degree $\delta$ either is hamiltonian or contains a cycle of length at least $2\delta$. In 1986, Bauer and Schmeichel enlarged the bound $2\delta$ to $2\delta+2$ under additional 1-tough condition - an alternative and more natural necessary condition for a graph to be hamiltonian. In fact, the bound $2\delta+2$ is sharp for a graph on $n$ vertices when $n\equiv 1(mod\ 3)$. In this paper we present the final version of this result which is sharp for each $n$: every 1-tough graph either is hamiltonian or contains a cycle of length at least $2\delta+2$ when $n\equiv 1(mod\ 3)$, at least $2\delta+3$ when $n\equiv 2(mod\ 3)$ or $n\equiv 1(mod\ 4)$, and at least $2\delta+4$ otherwise.