On The Binomial Edge Ideal of a Pair of Graphs (original) (raw)

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Keywords: Binomial edge ideal of a pair of graphs, Linear resolutions, Linear relations, Castelnuovo-Mumford regularity

Abstract

We characterize all pairs of graphs (G1,G2)(G_1,G_2)(G1,G2), for which the binomial edge ideal JG1,G2J_{G_1,G_2}JG1,G2 has linear relations. We show that JG1,G2J_{G_1,G_2}JG1,G2 has a linear resolution if and only if G1G_1G1 and G2G_2G2 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 (G1,G2)(G_1,G_2)(G1,G2) with girth (i.e. the length of a shortest cycle in the graph) greater than 3, betai,i+2(JG1,G2)=0\beta_{i,i+2}(J_{G_1,G_2})=0betai,i+2(JG1,G2)=0, for all iii. Moreover, we give a lower bound for the Castelnuovo-Mumford regularity of any binomial edge ideal JG1,G2J_{G_1,G_2}JG1,G2 and hence the ideal of adjacent 222-minors of a generic matrix. We also obtain an upper bound for the regularity of JG1,G2J_{G_1,G_2}JG_1,G2, if G1G_1G1 is complete and G2G_2G_2 is a closed graph.

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How to Cite

Saeedi Madani, S., & Kiani, D. (2013). On The Binomial Edge Ideal of a Pair of Graphs. The Electronic Journal of Combinatorics, 20(1), #P48. https://doi.org/10.37236/2987