Regularity of two classes of Cohen-Macaulay binomial edge ideals

TL;DR

This paper derives explicit combinatorial formulas for the Castelnuovo-Mumford regularity of two classes of Cohen-Macaulay binomial edge ideals, focusing on graphs with specific structures like chains of cycles and regular blocks with whiskers.

Contribution

It introduces new combinatorial invariants and formulas for the regularity of Cohen-Macaulay binomial edge ideals in specific graph classes, advancing classification methods.

Findings

Derived formulas for regularity of chain of cycles with whiskers.

Presented a linear formula for regularity of r-regular r-connected blocks with whiskers.

Introduced a new graph invariant related to block structures.

Abstract

Some recent investigations indicate that for the classification of Cohen-Macaulay binomial edge ideals, it suffices to consider biconnected graphs with some whiskers attached (in short, `block with whiskers'). This paper provides explicit combinatorial formulae for the Castelnuovo-Mumford regularity of two specific classes of Cohen-Macaulay binomial edge ideals: (i) chain of cycles with whiskers and (ii) $r$-regular $r$-connected block with whiskers. For the first type, we introduce a new invariant of graphs in terms of the number of blocks in certain induced block graphs, and this invariant may help determine the regularity of other classes of binomial edge ideals. For the second type, we present the formula as a linear function of $r$.