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Binomial Edge Ideals and Related Ideals | SpringerLink

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Binomial Ideals pp 171–238Cite as

Binomial Edge Ideals and Related Ideals

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Abstract

In this chapter we consider classes of binomial ideals which are naturally attached to finite simple graphs. The first of these classes are the binomial edge ideals. These ideals may also be viewed as ideals generated by a subset of 2-minors of a (2 × n)-matrix of indeterminates. Their Gröbner bases will be computed. Graphs whose binomial edge ideals have a quadratic Gröbner basis are called closed graphs. A full classification of closed graphs is given. For an arbitrary graph the initial ideal of the binomial edge ideal (for a suitable monomial order) is a squarefree monomial ideal. This has the pleasant consequence that the binomial edge ideal itself is a radical ideal. Its minimal prime ideals are determined in terms of cut point properties of the underlying graph. Based on this information, the closed graphs whose binomial edge ideal is Cohen–Macaulay are classified. In the subsequent sections, the resolution of binomial edge ideals is considered and a bound for the Castelnuovo–Mumford regularity of these ideals is given. Finally, the Koszul property of binomial edge ideals is studied. Intimately related to binomial edge ideals are permanental edge ideals and Lovász, Saks, and Schrijver edge ideals. Their primary decomposition will be studied.

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Authors and Affiliations

  1. Fakultät für Mathematik, Universität Duisburg-Essen, Essen, Germany

    Jürgen Herzog

  2. Department of Pure & Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Suita, Osaka, Japan

    Takayuki Hibi

  3. Department of Mathematical Sciences, School of Science and Technology, Kwansei Gakuin University, Sanda, Hyogo, Japan

    Hidefumi Ohsugi

Authors
  1. Jürgen Herzog
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  2. Takayuki Hibi
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  3. Hidefumi Ohsugi
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Herzog, J., Hibi, T., Ohsugi, H. (2018). Binomial Edge Ideals and Related Ideals. In: Binomial Ideals. Graduate Texts in Mathematics, vol 279. Springer, Cham. https://doi.org/10.1007/978-3-319-95349-6_7

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