Edge rings of bipartite graphs with linear resolutions. (English) Zbl 1476.05208
Summary: On the one hand, H. Ohsugi and T. Hibi [Adv. Appl. Math. 22, No. 1, 25–28 (1999; Zbl 0916.05046)] characterized the edge ring of a finite connected simple graph with a \(2\)-linear resolution. On the other hand, T. Hibi et al. [Proc. Am. Math. Soc. 147, No. 8, 3225–3232 (2019; Zbl 1416.05307)] conjectured that the edge ring of a finite connected simple graph with a \(q\)-linear resolution, where \(q\geq 3\), is a hypersurface and proved the case \(q=3\). In this paper, we solve this conjecture for the case of finite connected simple bipartite graphs.
MSC:
| 05E40 | Combinatorial aspects of commutative algebra |
| 13H10 | Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) |
| 52B20 | Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry) |
Keywords:
edge ring; linear resolution; regularity; \(h^{\ast}\)-polynomial; edge polytope; root polytopeReferences:
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