Toric ideals associated with gap-free graphs

In this paper we prove that every toric ideal associated with a gap-free graph G has a squarefree lexicographic initial ideal. Moreover, in the particular case when the complementary graph of G is chordal (i.e. when the edge ideal of G   has a linear resolution), we show that there exists a reduced Gröbner basis GG of the toric ideal of G   such that all the monomials in the support of GG are squarefree. Finally, we show (using work by Herzog and Hibi) that if I is a monomial ideal generated in degree 2, then I has a linear resolution if and only if all powers of I have linear quotients, thus extending a result by Herzog, Hibi and Zheng.