Monotonicity of regularity for powers of arbitrary graphs

Prove that for every simple graph G and every positive integer d, the Castelnuovo–Mumford regularity of the edge ideal quotient does not increase when passing from the d-th power graph to the (d+1)-st power graph; namely, establish that reg(R/I(G^d)) ≥ reg(R/I(G^{d+1})).

Background

The paper studies the Castelnuovo–Mumford regularity of edge ideals associated with graph powers. The authors observe that the induced matching number is weakly decreasing under graph powering, and that induced matching number provides a lower bound for the regularity of an edge ideal. Motivated by computational evidence, they conjecture that regularity itself is weakly decreasing under graph powering for every simple graph.

The paper proves this conjecture for forests and cycles. It also notes that the conjecture holds for block graphs, but does not establish it for arbitrary simple graphs; therefore the general monotonicity assertion remains unresolved in the paper.