Let \(R=\mathbb{K}[x_1,\ldots,x_n]\) be a polynomial ring over a field \(\mathbb{K}\) and let \(M\) be a finitely generated graded \(R\)-module which has a minimal graded free resolution \[ \cdots \longrightarrow F_k\longrightarrow \cdots\longrightarrow F_0\longrightarrow M\longrightarrow0. \] Also, let \(a_k\) be the maximum of the degrees of the generators of \(F_k\). The Castelnuovo-Mumford regularity is defined as the smallest \(r\in\mathbb{Z}\) such that \(r\geq a_k-k\) for all \(k\).
The authors study bounds on this notion in toric ideals of graphs. In their first main result, they study the case that a finite simple graph \(G\) has an induced subgraph \(H\) of the form \(H=K_{n_1,n_1}\sqcup\cdots\sqcup K_{n_t,n_t}\) where each \(n_i\geq2\), where \(K_{n_i,n_i}\) is the complete bipartite graph. In this case, they prove that \[ \text{reg}(I_G)\geq n_1+n_2+\cdots+n_t-(t-1). \] In their second main result, they study the Castelnuovo-Mumford regularity for chordal bipartite graph ideals. Let \(G\) be a chordal bipartite graph with bartition \(V=\{x_1,\ldots,x_n\}\cup\{y_1,\ldots,y_m\}\) and we consider the numbers \(r=\mid \{x_i\;: \;\deg(x_i)=1\}\mid\) and \(s=\mid \{y_j\;: \;\deg(x_j)=1\}\mid\). The authors prove that \[ \text{reg}(I_G)\leq \min (n-r,m-s). \] By using their techniques, they close the manuscript by giving a new combinatorial proof for the graded Betti numbers of the toric ideal \(I_G\), where the graph \(G\) is the complete bipartite graph \(K_{2,n}\).