Regularity of powers of edge ideals: from local properties to global bounds (Q2200860)

Let \(k\) be a field and \(S=k[x_1, \ldots, x_n]\) be the polynomial ring in \(n\) variables over \(k\). Suppose that \(M\) is a graded \(S\)-module with minimal free resolution \[0 \longrightarrow \cdots \longrightarrow \bigoplus_{j}S(-j)^{\beta_{1,j}(M)} \longrightarrow \bigoplus_{j}S(-j)^{\beta_{0,j}(M)} \longrightarrow M \longrightarrow 0.\] The Castelnuovo-Mumford regularity (or simply, regularity) of \(M\), denote by \(\mathrm{reg}(M)\), is defined as follows: \[\mathrm{reg}(M)=\max\{j-i|\ \beta_{i,j}(M)\neq0\}.\] There is a natural correspondence between quadratic square-free monomial ideals of \(S\) and finite simple graphs with vertex set \(V(G)=\{x_1, \ldots, x_n\}\). Indeed, to any graph \(G\), one associates its edge ideal \(I(G)\) which is generated by quadratic square-free monomials corresponding to edges of \(G\). A graph \(G\) is locally of regularity \(\leq r\) if for every vertex \(x\in V(G)\), the inequality \(\mathrm{reg}(I(G) :x)\leq r\) holds. In the paper under review, the authors study the regularity of powers of edge ideals. It is proven that for every graph \(G\) and any integer \(s\geq 1\), we have \(\mathrm{reg}(I(G)^s)\leq 2s+\beta(G)-1\), where \(\beta(G)\) denotes the size of the largest matching of \(G\). Also, it is shown that if \(G\) is gap-free and locally of regularity at most \(r-1\), then for every integer \(s\geq 1\), the inequality \(\mathrm{reg}(I(G)^s)\leq 2s+r-2\) holds. Moreover, the authors prove that if \(G\) is gap-free and locally of regularity at most \(2\), then \(\mathrm{reg}(I(G)^s)=2s\), for every integer \(s\geq 2\). This is a weaker version of a conjecture of \textit{E. Nevo} et al. [J. Algebr. Comb. 37, No. 2, 243--248 (2013; Zbl 1274.13028)].