Combinatorics of Castelnuovo-Mumford Regularity of Binomial Edge Ideals

TL;DR

This paper introduces a new combinatorial invariant for graphs and establishes its relationship with the Castelnuovo-Mumford regularity of binomial edge ideals, providing new characterizations and interpretations especially for block graphs.

Contribution

It defines the invariant (G) and proves its equality with the regularity minus one for block graphs, offering a combinatorial perspective on algebraic invariants.

Findings

(G)  the regularity minus one for block graphs

Equality (G) = reg(J_G) - 1 characterizes closed and bipartite Cohen-Macaulay graphs

Provides a combinatorial interpretation of regularity for block graphs

Abstract

Since the introduction of binomial edge ideals $J_{G}$ by Herzog et al. and independently Ohtani, there has been significant interest in relating algebraic invariants of the binomial edge ideal with combinatorial invariants of the underlying graph $G$. Here, we take up a question considered by Herzog and Rinaldo regarding Castelnuovo--Mumford regularity of block graphs. To this end, we introduce a new invariant $\nu(G)$ associated to any simple graph $G$, defined as the maximal total length of a certain collection of induced paths within $G$ subject to conditions on the induced subgraph. We prove that for any graph $G$, $\nu(G) \leq \text{reg}(J_{G})-1$, and that the length of a longest induced path of $G$ is less than or equal to $\nu(G)$; this refines an inequality of Matsuda and Murai. We then investigate the question: when is $\nu(G) = \text{reg}(J_{G})-1$? We prove that equality holds when $G$ is closed; this gives a new characterization of a result of Ene and Zarojanu, and when $G$ is bipartite and $J_{G}$ is Cohen-Macaulay; this gives a new characterization of a result of Jayanathan and Kumar. For a block graph $G$, we prove that $\nu(G)$ admits a combinatorial characterization independent of any auxiliary choices, and we prove that $\nu(G) = \text{reg}(J_{G})-1$. This gives $\text{reg}(J_{G})$ a combinatorial interpretation for block graphs, and thus answers the question of Herzog and Rinaldo.