On the Girth of Hamiltonian Weakly Pancyclic Graphs
A graph is called weakly pancyclic if it contains cycles of all lengths between its girth and circumference. In answer to a question of Erdos, we show that a Hamiltonian weakly-pancyclic graph of order n can have girth as large as about 2√n/log n. In contrast to this, we show that the existence of a cycle of length at most 2√n - 1 is already implied by the existence of just two long cycles, of lengths n and n - 1. Moreover we show that any graph, Hamiltonian or otherwise, which has n + c edges will have girth of order at most (n/c) log c. © 1997 John Wiley & Sons, Inc. J Graph Theory 26: 165 173, 1997.
Journal of Graph Theory
Bollobás, B., & Thomason, A. (1997). On the Girth of Hamiltonian Weakly Pancyclic Graphs. Journal of Graph Theory, 26 (3), 165-173. https://doi.org/10.1002/(SICI)1097-0118(199711)26:3<165::AID-JGT7>3.0.CO;2-P