Dirac proved that any graph with minimum vertex degree δ contains either a cycle of length at least 2δ or a Hamilton cycle. Motivated by this result, we characterize those graphs having no cycle longer than 2δ.
@article{arxiv.math/0307123,
title = {Graphs with no $2\delta + 1$ cycle},
author = {Galen E. Turner},
journal= {arXiv preprint arXiv:math/0307123},
year = {2007}
}