Castelnuovo-Mumford regularity of generalized binomial edge ideals of graphs

TL;DR

This work analyzes the Castelnuovo-Mumford regularity of generalized binomial edge ideals $J_{K_m,G}$, addressing whether all integers from $2$ to $n-1$ can be realized for graphs on $n$ vertices. It develops tight bounds for the regularity of join products $G=G_1*G_2$, showing reg values lie in the set $\{\mathrm{reg}(S/J_{K_m,G_1}),\mathrm{reg}(S/J_{K_m,G_2}),m-1,m\}$ and uses these to realize a wide range of regularities, including all $3\le r\le n-2$ via joins. The paper also provides a complete classification of graphs with reg $=2$, characterizes $P_4$-free graphs with a new join-decomposition perspective, and identifies extremal Gorenstein generalized binomial edge ideals, showing they occur only for $m=3$ and $G=K_3$. Together, these results illuminate when $S/J_{K_m,G}$ attains special homological properties (e.g., Cohen-Macaulay, Gorenstein) and spotlight several open structural questions for future work.

Abstract

In this paper, we mainly study the Castelnuovo-Mumford regularity of the generalized binomial edge ideals of graphs. We show that this number can be any integer number from $2$ to $n-1$ where $n$ is the number of vertices in the underlying graph. We are able to show this, after giving some tight lower and upper bounds for the regularity of generalized binomial edge ideals of the join product of graphs. In particular, we characterize all generalized binomial edge ideals with the regularity equal to~$2$ as well as extremal Gorenstein ideals. For this purpose, we give a new combinatorial characterization for the class of $P_4$-free graphs.

Theorems & Definitions (28)

• Definition 2.1
• Proposition 2.2
• Proposition 2.3
• Proposition 2.4
• Theorem 2.5
• Lemma 2.6
• Theorem 2.7
• proof
• Corollary 2.8
• proof