Venue
Online (Google Meet)
Speaker
Rimpa Nandi
Affiliation
SOA University, Bhubaneswar
Title
Castelnuovo-Mumford regularity and linear resolution property of ideals arising from graphs
Let $G$ be a simple finite graph with vertex set $V(G)=\{x_1,x_2,x_3,\dots,x_n\}$ and edge set $E(G)=\{e_1,e_2,e_3,\dots,e_q\}$. Also suppose that $I(G)$ is the edge ideal of $G$, where $I(G)=\langle x_{i}x_{j}~|~\{x_i,x_j\}\in E(G)\rangle$ $\subset R=K[x_1,x_2,\dots,x_n].$ We assume that $R(I(G))$ and $K[G]$ are the Rees algebra and toric algebra of $I(G)$ respectively.
In this talk we give a new upper bound for the regularity of edge ideals of gap-free graphs, in terms of their minimal triangulation. We also provide a new class of gap-free graphs such that $I(G)^s$ has linear resolution for $s\geq 3.$
We also show that if $G$ is connected and $R(I(G))$ is normal, then $\reg(R(I(G)))\leq \alpha_0(G)$, where $\alpha_0(G)$ is the vertex cover number of $G$. As a consequence, every normal K\"onig 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 the toric algebra $K[G]$ has $q$-linear resolution$(q\geq 4)$, then $K[G]$ is a hypersurface, that is the defining ideal $I_G$ of $K[G]$ is generated by a single element, which proves the conjecture of Hibi-Matsuda-Tsuchiya affirmatively for chordal graphs.