Minimum cut-sets and vertex connectivity in the unresolved composite-exponent regime

Characterize all minimum cut-sets and determine the vertex connectivity of the power graph P(C_n) of a finite cyclic group C_n when n has at least four distinct prime divisors, 2φ(p_1p_23p_{r-1})<p_1p_2p_{r-1}, and n_i>1 for at least one i[r].

Background

For n=p_1^{n_1}p_2^{n_2}p_r^{n_r}, the paper studies minimum cut-sets and vertex connectivity in the power graph P(C_n), where p_1<<p_r are the distinct prime divisors of n. Prior work had reduced the possible minimum cut-sets to specific families, including the sets Z_a^s and X_{a,b}^{s,t}, but had not completely resolved their relative sizes or determined which candidates are minimum in all parameter regimes.

The paper proves substantial results when n_r2 and when r{4,5} with n_r=1. However, Theorem 1.2 leaves a genuine alternative between Z_r^{n_r} and Z_b^{n_b} when n_r=2 and the set is nonempty; such cases can occur for r=6, as illustrated in Example 6.1. Thus the stated general characterization and connectivity problem remains unresolved beyond the cases settled by the paper.