Minimum cut-sets and vertex connectivity in the remaining cyclic power-graph cases

Characterize the 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_2\cdots p_{r-1})<p_1p_2\cdots p_{r-1}, and at least one exponent n_i is greater than 1.

Background

For n=p_1^{n_1}p_2^{n_2}\cdots p_r^{n_r}, the paper studies minimum cut-sets and vertex connectivity in the power graph P(C_n), where the primes satisfy p_1<p_2<\cdots<p_r. Prior work had identified possible forms of minimum cut-sets and established several cases of the vertex-connectivity problem, but had not completely characterized all possibilities when r≥4.

The explicitly identified unresolved regime is constrained by the inequality 2φ(p_1p_2\cdots p_{r-1})<p_1p_2\cdots p_{r-1} and by the requirement that at least one prime exponent n_i exceed 1. The present paper resolves important subcases, including all cases with n_r≥3 or with r∈{4,5} under the stated hypotheses, while its theorem for n_r=2 and nonempty Ω leaves alternative candidates whose exact minimum still requires determination.