In the talk we shall present some recent results, obtained jointly with Andrew Thomason. A graph is called weakly pancyclic if it contains cycles of all lengths between its girth and circumference. A substantial result of Haggkvist, Faudree and Schelp (1981) states that a Hamiltonian non-bipartite graph of order n and size at least
contains cycles of every length between 3 and n. From this, Brandt (1996) 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
One of our main results is a slightly weaker form of this conjecture, namely that every graph of order n and size at least
is weakly pancyclic or bipartite.