Let be a simple graph. We show that if is connected and is normal, , where is the vertex cover number of . As a consequence, for every normal König connected graph , , the matching number of . For a gap-free graph , we give various combinatorial upper bounds for . As a consequence we give various sufficient conditions for the equality of and . Finally we show that if is a chordal graph such that has -linear resolution (), then is a hypersurface, which proves the conjecture of Hibi, Matsuda and Tsuchiya [10, Conjecture 0.2] affirmatively for chordal graphs.
"On regularity bounds and linear resolutions of toric algebras of graphs." J. Commut. Algebra 14 (2) 285 - 296, Summer 2022. https://doi.org/10.1216/jca.2022.14.285