Binomial arithmetical rank of edge ideals of forests (Q2838950)

Let \(R=\mathbb{K}[x_1,\dots,x_n]\) be a polynomial ring over an arbitrary field \(\mathbb{K}\) and let \(I\) be a squarefree monomial ideal of \(R\). In the ideal \(I\) the following notions are known: NEWLINE{\parindent=6mmNEWLINE\begin{itemize}\item[--]The arithmetical rank of I is denoted by \(\mathrm{ara}I\) and is defined as the minimum number \(k\) of elements \(a_1,\dots,a_k \in R\) such that \(\sqrt{<a_1,\dots,a_k>}=I\). NEWLINE\item[--]The binomial arithmetic rank of \(I\) is denoted by \(\mathrm{biara}I\) and is defined as the minimum number \(k\) of binomials or monomials of \(R\) such that \(\sqrt{<a_1,\dots,a_k>}=I\). \item[--]The big height of \(I\) is denoted by \(\mathrm{bight}I\) and is defined as the maximum height of the minimal prime ideals of \(I\). NEWLINE\item[--]Finally the projective dimension of the quotient ring \(R/I\) is defined by \(\mathrm{pd}(R/I)\). NEWLINENEWLINE\end{itemize}} NEWLINEIt is remarked that the following inequalities hold: NEWLINE\[NEWLINE\mathrm{bight}I\leq \mathrm{pd}(R/I)\leq \mathrm{ara}I\leq \mathrm{biara}I.NEWLINE\]NEWLINE NEWLINEThe authors are interested in studying when the equality \(\mathrm{ara}I=\mathrm{pd}(R/I)\) holds for the special case of edge ideals \(I_G\), where \(G\) is a forest, i.e., graphs without cycles.NEWLINENEWLINEPrecisely in their main theorem they are proving for a forest \(G\) that NEWLINE\[NEWLINE\mathrm{bight}I_G=\mathrm{biara}I_G.NEWLINE\]NEWLINE In order to obtain that they use as basic key tools graph theory and especially tree like systems and primitive trees of the forest \(G\) which are connected by the authors with the minimal vertex coverings of the trees of \(G\).