Characterization of Some Graphs Realizing Regularity Bounds for Binomial Edge Ideals

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 $[\ell(G), |V(G)|-ω(G)+1]$.

💡 Research Summary

The paper investigates the Castelnuovo‑Mumford regularity of binomial edge ideals (J_G) associated to a simple graph (G). For a graph on (n) vertices, the regularity (\operatorname{reg}(S/J_G)) is known to satisfy several bounds: the lower bound (\ell(G)), which is the sum of the lengths of the longest induced paths in each connected component; the upper bound (c(G)), the number of maximal cliques; and another upper bound (n-\omega(G)+1), where (\omega(G)) is the size of a largest clique. While each bound can be sharp for particular families, the authors focus on the situation where two of these bounds coincide, i.e., when the regularity simultaneously attains both a lower and an upper bound.

The first major contribution is a complete characterization of graphs for which (\operatorname{reg}(S/J_G)=\ell(G)=c(G)). Earlier work (Malayeri et al.) proved this equality for chordal graphs, calling them “strongly interval graphs”. The present work removes the chordal restriction and introduces a broader class called CL‑graphs (Connected‑CL‑graphs). A CL‑graph is defined via an intersection graph built from a family of intervals ({J_0,J_1,\dots,J_\ell}) representing a longest induced path and additional sets ({I_1,\dots,I_r}) that are unions of disjoint sub‑intervals satisfying specific separation and Helly‑type conditions. The authors prove that a graph satisfies (\ell(G)=c(G)) if and only if each of its connected components is isomorphic to such an intersection graph, i.e., it is a CL‑graph. Consequently, for CL‑graphs the regularity equals the lower bound (\ell(G)) and also the number of maximal cliques.

The second main result deals with connected graphs where the regularity meets the bound (|V(G)|-\omega(G)+1). The authors define WL‑graphs (Weak‑Lattice graphs) as graphs that consist of a single induced path together with a sequence of pairwise disjoint cliques (complete subgraphs) that “cover” consecutive blocks of the path. In a WL‑graph each maximal clique corresponds to a block of consecutive vertices on the path, and the length of the path plus the number of cliques is exactly (|V(G)|-\omega(G)+1). They prove that a connected graph satisfies (\operatorname{reg}(S/J_G)=\ell(G)=|V(G)|-\omega(G)+1) precisely when it is a WL‑graph. Thus WL‑graphs give a full description of the graphs attaining the second pair of bounds.

Beyond the characterizations, the paper addresses the possible values of regularity inside the intervals (