Asymptotic regularity of invariant chains of edge ideals

We study chains of nonzero edge ideals that are invariant under the action of the monoid $\mathrm{Inc}$ of increasing functions on the positive integers. We prove that the sequence of Castelnuovo--Mumford regularity of ideals in such a chain is eventually constant with limit either 2 or 3, and we determine explicitly when the constancy behaviour sets in. This provides further evidence to a conjecture on the asymptotic linearity of the regularity of $\mathrm{Inc}$-invariant chains of homogeneous ideals. The proofs reveal unexpected combinatorial properties of $\mathrm{Inc}$-invariant chains of edge ideals.