Betti numbers of toric ideals of graphs: A case study

We compute the graded Betti numbers for the toric ideal of a family of graphs constructed by adjoining a cycle to a complete bipartite graph. The key observation is that this family admits an initial ideal which has linear quotients. As a corollary, we compute the Hilbert series and $h$-vector for all the toric ideals of graphs in this family.
Comment: 12 pages; revised and corrected; to appear in Journal of Algebra and its Applications