A Weakly Pancyclic Theorem for Hamiltonian Non-Bipartite Graphs
Hamilton非二部图的弱泛圈性

An n-vertex graph is called weakly pancyclic if it contains cycles of all lengths between its girth and circumference. In 1977, Brandt conjectured that an n-vertex non-bipartite graph with more than \lfloor {{\textstyle{{n^2 } \over 4}}}\rfloor- n + 5 edges is weakly pancyclic. Bollobas and Thomason(1999) proved that every non-bipartite graph of order n and size at least \lfloor{{{\textstyle{{n^2 } \over 4}}}\rfloor - n + 59 is weakly pancyclic. In this paper, the following result is established: let G be a Hamiltonian non-bipartite graph of order $n$ and size at least \lfloor {{\textstyle{{n^2 } \over 4}}}\rfloor - n + 12, then G is weakly pancyclic.