Regularity 3 in edge ideals associated to bipartite graphs
Regularity 3 in edge ideals associated to bipartite graphs
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与二分图相关的边理想中的正则 3
DOI：
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发表时间：
2012
期刊：
影响因子：
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通讯作者：
P. Gimenez
中科院分区：
文献类型：
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作者：
Oscar Fernández;P. Gimenez
在本文中，我们专注于与二部图的边理想，并给出了那些具有正则性3的组合特征。当正则性严格大于3时，我们确定了最小分次自由归结的第一步i，其中存在度>i+3的最小生成元，证明了在这一步最小生成元的最高度为i+4，并利用图的组合学确定了相应的分次Betti数βi，i+4.然后，通过极化将结果推广到非无平方的情况。我们还研究了一族正则性4的理想，它们在我们的主要结果中发挥着重要作用，并且其分次Betti数可以通过封闭的组合公式完全描述。
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 βi,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.