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Normality of cover ideals of graphs and normality under some operations. (English) Zbl 1422.13009
Authors’ abstract: We produce a procedure for constructing new normal monomial ideals from other ideals that are assumed to be normal. This enables us to prove that if the cover ideal of a graph \(G\) is normal, then the cover ideal of the graph \(H\) is normal as well, where the graph \(H\) is obtained by connecting all vertices in \(G\) with a new vertex. We use these ideas to explore the normality of the cover ideals of some imperfect graphs. Also, we investigate the normality under expansion, this leads us to generalize the work of I. Al-Ayyoub [Rocky Mt. J. Math. 39, No. 1, 1–9 (2009; Zbl 1166.13023)]. Furthermore, we investigate the normality under more operations such as weighting, polarization, localization, contraction, and deletion.
MSC:
13B22 Integral closure of commutative rings and ideals
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
05E40 Combinatorial aspects of commutative algebra
Software:
Macaulay2; Normaliz
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References:
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