k-deep connectivity of the (k+1)-core

Determine whether the (k+1)-core of the random graph is k-deeply connected with high probability.

Background

A graph is defined to be k-deeply connected when it contains k edge-disjoint spanning trees. The paper proves structural results for giant k-deeply connected components and separately analyzes the (k+1)-core through its size and rank.

Although the paper establishes a close asymptotic relationship between these structures, it does not prove that the entire (k+1)-core itself satisfies k-deep connectivity. This property is explicitly posed as an open question.