On regularity bounds and linear resolutions of toric algebras of graphs

Let G be a simple graph. In this article we show that if G is connected and R(I(G)) is normal, then reg(R(I(G))) ≤ α0(G), where α0(G) the vertex cover number of G. As a consequence, 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-Tsuchiya [12, Conjecture 0.2], affirmatively for chordal graphs.