Graphs and Ideals Generated by Some 2-Minors

Abstract

Let G be a finite graph on [n] = {1, 2,…, n}, X a 2 × n matrix of indeterminates over a field K, and S = K[X] a polynomial ring over K. In this article, we study about ideals I _G of S generated by 2-minors [i, j] of X which correspond to edges {i, j} of G. In particular, we construct a Gröbner basis of I _G as a set of paths of G and compute a primary decomposition.

Key Words:

• Gröbner basis
• Hamilton cycle
• 2-Minor
• Primary decomposition

2000 Mathematics Subject Classification:

• Primary 13P10
• Secondary 13C15

ACKNOWLEDGMENTS

The author is grateful to Mitsuyasu Hashimoto, Yuhi Sekiya and Ken-ichi Yoshida for valuable conversations and helpful suggestions. He also expresses his thanks to the referee for his many pieces of valuable advice. In particular, the example in Remark 5.6 is due to him.

The author was partially supported by JSPS Research Fellowships for Young Scientists.

Notes

After the first version of this article was written, very recently, it has been brought to my attention that Herzog, Hibi, Hreinsdóttir, Kahle, and Rauh wrote an article [6] which has considerable overlaps with this article.

Communicated by S. Goto.