On the binomial edge ideal of a pair of graphs
Let \(G_{1}\) be a graph on the vertex set \([m]\) and \(G_{2}\) a graph on the vertex set \([n]\), and let \(X=(x_{ij})\) be an\((m\times n)\)-matrix of indeterminates. Let \(K[X]\) be the polynomial ring over a field \(K\) in the variables \(x_{ij}\), where \(i=1,\dots ,m\) and \(j=1,\dots ,n\). Let \(e=\{i,j\}\) for some \(1\leq i<j\leq m\) and \(f=\{t,l\}\) for some \(1\leq t<l\leq n.\) To the pair \((e,f)\), the following \(2\)-minor of \(X\) is assigned: \(p_{e,f}=[i,j|t,l]=x_{it}x_{jl}-x_{il}x_{jt}\). Then the ideal \(J_{G_{1},G_{2}}=(p_{e,f}:e\in E(G_{1}),f\in E(G_{2}))\) is called the binomial edge ideal of the pair \((G_{1},G_{2})\). In this paper, the authors study some algebraic properties and invariants of \(J_{G_{1},G_{2}}\). In Theorem 1, the authors show the following conditions are equivalent: (a) \(J_{G_{1},G_{2}}\) has linear relations; (b) \(J_{G_{1},G_{2}}\) is a toric ideal and \(G_{1}\) and \(G_{2}\) are connected; (c) \(G_{1}\) and \(G_{2}\) are complete graphs. In Theorem 10, the authors further show that \(J_{G_{1},G_{2}}\) has a linear resolution if and only if \(G_{1}\) and \(G_{2}\) are complete graphs and one of them is just an edge graph. Let \(G\) be a finite simple graph with vertex set \([n]\) and edge set \(E(G)\). Also, let \(S=K[x_{1},\dots ,x_{n},y_{1},\dots ,y_{n}]\) be the polynomial ring over a field \(K\). Then the binomial edge ideal of \(G\) in \(S\), denoted by \(J_{G}\), is generated by binomials of the form \(f_{ij}=x_{i}y_{j}-x_{j}y_{i}\) where \(i<j\) and \(\{i,j\}\in E(G)\). When \(J_{G}\) has a quadratic Gröbner basis with respect to the lexicographic order induced by \(x_{1}>\dots>x_{n}>y_{1}>\dots>y_{n}\), \(G\) is called a closed graph. Let \(e(G)\) be the number of edges of \(G\), \(c(G)\) be the number of maximal cliques of \(G\), \(K_{m}\) be the complete graph on \([m]\). In Theorem 13, the authors show that when \(G\) is a closed graph then the Castelnuovo-Mumford regularity of \(J_{K_{m},G}\) is bounded above by \(\min\{\binom{m}{2}c(G),e(G)\}+1\).
scientific article; zbMATH DE number 6171887