Weakly Pancyclic Graphs
A graph is called weakly pancyclic if it contains cycles of all lengths between its girth and circumference. A substantial result of Häggkvist, Faudree, and Schelp (1981) states that a Hamiltonian non-bipartite graph of order n and size at least ⌊(n-1)2/4⌋+2 contains cycles of every length l, 3≤l≤n. From this, Brandt (1997) deduced that every non-bipartite graph of the stated order and size is weakly pancyclic. He conjectured the much stronger assertion that it suffices to demand that the size be at least ⌈n2/4⌉-n+5. We almost prove this conjecture by establishing that every graph of order n and size at least ⌊n2/4⌋-n+59 is weakly pancyclic or bipartite. © 1999 Academic Press.
Journal of Combinatorial Theory. Series B
Bollobás, B., & Thomason, A. (1999). Weakly Pancyclic Graphs. Journal of Combinatorial Theory. Series B, 77 (1), 121-137. https://doi.org/10.1006/jctb.1999.1916