Geodesic
Regularity 3 in edge ideals associated to bipartite graphs
Journal of Algebraic Combinatorics, Tome 39 (2014) no. 4, pp. 919-937.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

We focus in this paper on edge ideals associated to bipartite graphs and give a combinatorial characterization of those having regularity 3. When the regularity is strictly bigger than 3, we determine the first step $i$ in the minimal graded free resolution where there exists a minimal generator of degree $>i+3$, show that at this step the highest degree of a minimal generator is $i+4$, and determine the corresponding graded Betti number $\beta_{i,i+4}$ in terms of the combinatorics of the graph. The results are then extended to the non-square-free case through polarization. We also study a family of ideals of regularity 4 that play an important role in our main result and whose graded Betti numbers can be completely described through closed combinatorial formulas.
Classification : 05C25
Keywords: edge ideal, bipartite graph, Castelnuovo-Mumford regularity, graded Betti numbers, independence complex, Stanley-Reisner ideal 
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     title = {Regularity 3 in edge ideals associated to bipartite graphs},
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Fernández-Ramos, Oscar; Gimenez, Philippe. Regularity 3 in edge ideals associated to bipartite graphs. Journal of Algebraic Combinatorics, Tome 39 (2014) no. 4, pp. 919-937. http://geodesic.mathdoc.fr/item/JAC_2014__39_4_a2/