On regularity bounds and linear resolutions of toric algebras of graphs

Abstract

Let G be a simple graph. We show that if G is connected and R(I (G)) is normal, reg ⁡(R(I (G)))≤α0 (G), where α0 (G) is the vertex cover number of G. As a consequence, for every normal König connected graph G, reg ⁡(R(I (G)))= mat ⁡ (G), the matching number of G. For a gap-free graph G, we give various combinatorial upper bounds for reg ⁡(R(I (G))). As a consequence we give various sufficient conditions for the equality of reg ⁡(R(I (G))) and mat ⁡ (G). Finally we show that if G is a chordal graph such that K[G] has q-linear resolution (q≥4), then K[G] is a hypersurface, which proves the conjecture of Hibi, Matsuda and Tsuchiya [10, Conjecture 0.2] affirmatively for chordal graphs.