Logarithmic Circumference In Tough Grahphs

For every real $t>0$, we prove that every $2$-connected $t$-tough graph contains a cycle of length at least $\mathrm{\ell }$ whenever $n\le \mathrm{\ell }(1+1\mathrm{/}t{)}^{\lfloor \mathrm{\ell }\mathrm{/}2\rfloor -1}$. This establishes the logarithmic bound conjectured by Broersma, van den Heuvel, Jung, and Veldman.