Regularity of powers of forests and cycles

TL;DR

This paper explicitly computes the regularity of powers of edge ideals for forests and cycles, providing exact formulas and bounds, and explores asymptotic linearity and initial thresholds for these classes of graphs.

Contribution

It offers explicit formulas for the regularity of I^s for forests and cycles, and establishes new bounds for graphs with Hamiltonian paths or cycles.

Findings

Computed regularity of I^s for all s > 0 in forests and cycles

Established asymptotic linearity of reg(I^s) for these graph classes

Provided bounds on regularity for Hamiltonian graphs

Abstract

Let G be a graph and let I = I(G) be its edge ideal. In this paper, when G is a forest or a cycle, we explicitly compute the regularity of I^s for all s > 0. In particular, for these classes of graphs, we provide the asymptotic linear function reg(I^s) as s > 0, and the initial value of s starting from which reg(I^s) attains its linear form. We also give new bounds on the regularity of I when G contains a Hamiltonian path and when G is a Hamiltonian graph.