Combinatorics of Castelnuovo-Mumford Regularity of Binomial Edge Ideals

TL;DR

This work introduces the invariant ν(G) for simple graphs and proves the fundamental bound ν(G) ≤ reg(S/J_G) − 1, while also showing that the longest induced path length ell(G) ≤ ν(G). It establishes valuable equality outcomes: for closed graphs, reg(S/J_G) = ell(G) = ν(G); for Cohen–Macaulay bipartite graphs, reg(S/J_G) = ν(G). The centerpiece is a complete combinatorial characterization for block graphs, proving ν(G) = reg(S/J_G) and giving a labeling-free description via forbidden subgraphs in P_Ind, thereby providing a concrete combinatorial interpretation of regularity in this important graph class. Overall, ν(G) serves as a unified framework tying algebraic invariants of binomial edge ideals to precise combinatorial structures, resolving Herzog and Rinaldo’s question for block graphs and extending known cases.

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 $ν(G)$ 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 $G$, $ν(G) \leq \text{reg}(J_{G})-1$, and that the length of a longest induced path of $G$ is less than or equal to $ν(G)$; this refines an inequality of Matsuda and Murai. We then investigate the question: when is $ν(G) = \text{reg}(J_{G})-1$? 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 $ν(G)$ admits a combinatorial characterization independent of any auxiliary choices, and we prove that $ν(G) = \text{reg}(J_{G})-1$. This gives $\text{reg}(J_{G})$ a combinatorial interpretation for block graphs, and thus answers the question of Herzog and Rinaldo.

Figures (10)

• Figure 1: DOIP Paths
• Figure 2: Non-DOIP paths
• Figure 3: Dependence of $K_{P_{\mathop{\mathrm{Ind}}\nolimits}}$ on Orientations
• Figure 4: $F_{m}$, $m \leq 4$
• Figure 6: Block Graph

Theorems & Definitions (108)

• Theorem A
• Lemma 2.1
• Definition 2.2
• Definition 2.3
• Lemma 2.4: thulasiraman2011graphs
• Definition 2.5: herzog2010binomial
• Definition 2.6: herzog2010binomial
• Remark 2.7
• Theorem 2.8: herzog2010binomial
• Theorem 2.9: conca2020square