Upper bounds for the regularity of gap-free graphs in terms of minimal triangulation (Q2114466)

Let \(\mathbb{K}\) be a field and \(R=\mathbb{K}[x_1,\ldots,x_n]\) be the polynomial ring in \(n\) variables over \(\mathbb{K}\). Suppose that \(M\) is a nonzero graded \(R\)-module with minimal free resolution \[0 \longrightarrow \cdots \longrightarrow \bigoplus_{j}R(-j)^{\beta_{1,j}(M)} \longrightarrow \bigoplus_{j}R(-j)^{\beta_{0,j}(M)} \longrightarrow M \longrightarrow 0.\] The Castelnuovo-Mumford regularity (or simply, regularity) of \(M\), denoted by \(\mathrm{reg}(M)\), is defined as \[\mathrm{reg}(M)=\max\{j-i\mid \beta _{i,j}(M)\neq 0\}.\] Let \(G\) be a finite simple graph with vertex set \(V(G)=\big\{x_1, \ldots, x_n\big\}\) and edge set \(E(G)\). The edge ideal \(I(G)\) of \(G\) defined by \[I(G)=\big(x_ix_j: \{x_i,x_j\}\in E(G)\big).\] Two disjoint edges \(e, e'\in E(G)\) are said to be a gap, if they form an induced subgraph of \(G\). The graph \(G\) is gap-free if it has no gap. In the paper under review, the authors provide an upper bound for the regularity of edge ideals of gap-free graphs, in terms of their minimal triangulation. It is shown that \(\mathrm{reg}(I(G))\leq \mathrm{reg}(I(\mathcal{C}_U))\), where \(\mathcal{C}_U\) is a \(3\)-uniform clutter which consists of certain \(3\)-paths in a minimal triangulation of \(G\). By using this result, a general upper bound for the regularity of gap-free graphs is studied. If \(\mathcal{H}\) is the \(3\)-uniform clutter which consists of certain \(3\)-cliques in \(G\) or in its triangulation and \(\mathcal{H}\) is chordal, then \(\mathrm{reg}(I(G))\leq 3\).