Algebraic properties of toric rings of graphs

Abstract

Let $G=(V,E)$ be a simple graph. We investigate the Cohen–Macaulayness and algebraic invariants, such as the Castelnuovo-Mumford regularity and the projective dimension, of the toric ring k[G] via those of toric rings associated to induced subgraphs of G.

Keywords:

• Cohen–Macaulay
• graphs
• odd cycle condition
• projective dimension
• regularity
• toric rings

2000 MATHEMATICS SUBJECT CLASSIFICATION:

• 13D02
• 13F55
• 14M25

Acknowledgements

Part of this work was done when the first two authors were visiting Vietnam Institute for Advanced Study in Mathematics (VIASM). The authors would like to thank VIASM for its support and hospitality. The authors would also like to thank K. Kimura for providing Example 5.4 and for spotting a gap in our first draft.

Additional information

Funding

The first named author is partially supported by the Simons Foundation (grant #279786).