On the binomial edge ideal of a pair of graphs

Sara Saeedi Madani¹ ; Dariush Kiani¹

¹Amirkabir University

The electronic journal of combinatorics, Tome 20 (2013) no. 1

Résumé

We characterize all pairs of graphs $(G_1,G_2)$, for which the binomial edge ideal $J_{G_1,G_2}$ has linear relations. We show that $J_{G_1,G_2}$ has a linear resolution if and only if $G_1$ and $G_2$ are complete and one of them is just an edge. We also compute some of the graded Betti numbers of the binomial edge ideal of a pair of graphs with respect to some graphical terms. In particular, we show that for every pair of graphs $(G_1,G_2)$ with girth (i.e. the length of a shortest cycle in the graph) greater than 3, $\beta_{i,i+2}(J_{G_1,G_2})=0$, for all $i$. Moreover, we give a lower bound for the Castelnuovo-Mumford regularity of any binomial edge ideal $J_{G_1,G_2}$ and hence the ideal of adjacent $2$-minors of a generic matrix. We also obtain an upper bound for the regularity of $J_{G_1,G_2}$, if $G_1$ is complete and $G_2$ is a closed graph.

Détail

DOI : 10.37236/2987

Classification : 13C05, 16E05, 05E40
Mots-clés : binomial edge ideal of a pair of graphs, linear resolutions, linear relations, Castelnuovo-Mumford regularity