The arithmetical rank of the edge ideals of graphs with pairwise disjoint cycles (Q2816882)

Let \(R\) be a unitary commutative Noetherian ring and \(I\) be an ideal of \(R\). The arithmetical rank of \(I\) is denoted by \(\mathrm{ara}(I)\) and is defined as the smallest integer \(t\) for which there exist \(a_1,\dots,a_t\in R\) such that \(\sqrt{\langle a_1, \dots,a_t\rangle}=\sqrt{I}\). \textit{G. Lyubeznik} [J. Pure Appl. Algebra 51, No. 1--2, 193--195 (1988; Zbl 0652.13012)] has proven that for every squarefree monomial \(I\), the inequality \(\mathrm{pd}_R(R/I)\leq\mathrm{ara}(I)\) holds. Therefore, we have the following chain of inequalities for every squarefree monomial ideal \(I\): NEWLINE\[NEWLINE\mathrm{ht}(I)\leq\mathrm{bight}(I)\leq\mathrm{pd}_R(R/I)\leq \mathrm{ara}(I).NEWLINE\]NEWLINENEWLINENEWLINEIn the paper under review, the authors study the difference of \(\mathrm{bight}(I)\) and \(\mathrm{ara}(I)\). More precisely, it is shown that if \(I\) is the edge ideal of a graph whose cycles are pairwise vertex-disjoint, then the difference of \(\mathrm{bight}(I)\) and \(\mathrm{ara}(I)\) is bounded above by the number of cycles.