Coincidence of the giant-component emergence time and the core threshold event

Determine whether the emergence time of the giant k-deeply connected component in the random graph process is, with high probability, the first time at which the (k+1)-core has v vertices and at least k(v-1) edges.

Background

The random graph process is formed by adding the edges of a complete graph in a uniformly random order. The paper defines the emergence time of the giant k-deeply connected component as the first process time at which that component appears.

The unresolved question asks whether this emergence event coincides with the first appearance of a (k+1)-core satisfying the edge-count condition k(v-1), where v is the number of vertices in the core. Establishing such a coincidence would connect the dynamic emergence of deep connectivity with a precise structural threshold for the core.

References

Let $\tau$ denote the moment of emergence of the giant $k$-deeply connected component in the process $G_1\subsetneq\dots\subsetneq G_{\binom{n}{2}$. Is that true that $\tau$ is w.h.p. also the first moment when the $(k+1)$ core has $v$ vertices and at least $k(v-1)$ edges?

— On the random minimum edge-disjoint spanning trees problem  (2502.08462 - Shabanov et al., 12 Feb 2025) in Section “Conclusion and Prospects”