Castelnuovo-Mumford regularity under reduction processes on Graphs and Hypergraphs

Abstract

In this paper, we introduce some reduction processes on graphs and hypergraphs which preserve the regularity of related circuit ideals. By these operations, we transform a (hyper)graph G to a smaller (hyper)graph G′ while the Castelnuovo-Mumford regularity does not change under these operations. As consequences of these reductions, we give an alternative proof for Fröbergʼs theorem [R. Fröberg, On Stanley-Reisner rings, in: Topics in algebra, Banach Center Publications, 26 Part 2, (1990), 57-70] on linearity of circuit ideals of chordal graph, as well as, linearity of circuit ideals of generalized chordal graphs as defined in [E. Emtander, A class of hypergraphs that generalizes chordal graphs, Math. Scand. 106 (2010), no. 1, 5066]. Moreover, a large class of hypergraphs such that their circuit ideals have a linear resolution is introduced. Finally, a formula for regularity of circuit ideals of decomposable hypergraphs will be given.