Characterization of Some Graphs Realizing Regularity Bounds for Binomial Edge Ideals
Nursel Erey, Muhammed Ergen, Takayuki Hibi
TL;DR
The paper studies the Castelnuovo-Mumford regularity $\operatorname{reg}(S/J_G)$ of binomial edge ideals, bounded below by $\ell(G)$ and above by $c(G)$ and by $|V(G)|-\omega(G)+1$, and seeks when these bounds coincide. It introduces CL-graphs to characterize graphs achieving $\operatorname{reg}(S/J_G)=\ell(G)=c(G)$ and WL-graphs for the case $\operatorname{reg}(S/J_G)=\ell(G)=|V(G)|-\omega(G)+1$, establishing exact equivalences and structural descriptions. It then proves broad realizability results: for any $2\le \ell \le r \le c$ there exists a connected graph with $\ell(G)=\ell$, $\operatorname{reg}(S/J_G)=r$, and $c(G)=c$, and for any $3\le \ell \le r \le \overline{\omega}$ there is a connected graph with $\ell(G)=\ell$, $\operatorname{reg}(S/J_G)=r$, and $|V(G)|-\omega(G)+1=\overline{\omega}$, using explicit constructions and gluing techniques. These results map the full range of feasible regularity values between the combinatorial bounds and provide a clear structural framework for extremal cases in binomial edge ideals.
Abstract
In this paper, we characterize all graphs $G$ satisfying \[\operatorname{reg}(S/J_G)=\ell(G)=c(G)\] where $\ell(G)$ is the sum of the lengths of the longest induced paths in each connected component of $G$ and $c(G)$ is the number of the maximal cliques of $G$. We also characterize all connected graphs $G$ that satisfy \[\operatorname{reg}(S/J_G)=\ell(G)=|V(G)|-ω(G)+1\] where $ω(G)$ is the clique number of $G$. Moreover, we investigate the possible values of the regularity of $S/J_G$ within the intervals $[\ell(G), c(G)]$ and $[\ell(G), |V(G)|-ω(G)+1]$.