Implosive graphs: Square-free monomials on symbolic Rees algebras

摘要

Let I be the edge monomial ideal of a graph G, whose vertex set is V(G) = {v(1),...,v(n)}. G is implosive if the symbolic Rees algebra R-S(I) of I has a minimal system of generators x(w1) t(b1),...,x(ws) t(bs) where x(w1),..., x(ws) are square-free monomials. We give some structural properties of implosive graphs and we prove that they are closed under cliquesums and odd subdivisions. Furthermore, we prove that universally signable graphs are implosive. We show that odd holes, odd antiholes and some Truemper configurations (prisms, thetas and even wheels) are implosive. Moreover, we study excluded families of subgraphs for the class of implosive graphs. In particular, we characterize which Truemper configurations and extensions of odd holes and antiholes are minimal nonimplosive.