By a conjecture of N. C. Minh, for any graph G and every integer s ≥ 1, one has the equality reg(I(G) (s) ) = reg(I(G) s ) (see [1]). If G is a bipartite graph, this conjecture is trivially true. The next class of graphs close to bipartite graphs that satisfies the conjecture are unicyclic graphs, i.e. the graphs with exactly one cycle (see [5]). To proceed, it is natural to ask this question for the family of locally bipartite graphs. One goal of this project is to study reg(I(G) (s) ) for a locally bipartite graph G and show the equality reg(I(G) (s) ) = reg(I(G) s ) for any s ≥ 1.
