On The Binomial Edge Ideal of a Pair of Graphs

• Published in: The Electronic journal of combinatorics 2013-03, Vol.20 (1)
• Main Authors: Saeedi Madani, Sara, Kiani, Dariush
• Format: Article
• Language: English
• ISSN: 1077-8926; 1077-8926
• DOI: 10.37236/2987

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.