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, $\mathrm{reg}(R(I(G)))\le {\alpha }_0(G)$, where ${\alpha }_0(G)$ is the vertex cover number of $G$. As a consequence, for every normal König connected graph $G$, $\mathrm{reg}(R(I(G)))=\mathrm{mat}(G)$, the matching number of $G$. For a gap-free graph $G$, we give various combinatorial upper bounds for $\mathrm{reg}(R(I(G)))$. As a consequence we give various sufficient conditions for the equality of $\mathrm{reg}(R(I(G)))$ and $\mathrm{mat}(G)$. Finally we show that if $G$ is a chordal graph such that $K[G]$ has $q$-linear resolution ($q\ge 4$), then $K[G]$ is a hypersurface, which proves the conjecture of Hibi, Matsuda and Tsuchiya [10, Conjecture 0.2] affirmatively for chordal graphs.