Existence of remainder-two colorings in unresolved generalized Petersen cases

Determine whether a closed coloring with remainder 2 modulo n exists for the generalized Petersen graph G(m,j) when 8 divides n, 4 divides m, and j is even.

Background

For remainder 2, the paper classifies several parity and divisibility regimes, including existence when j is odd and a complete criterion when 8|n, 2|m, 4∤m, and 2|j. One case remains unresolved: n is divisible by 8, m is divisible by 4, and j is even.

This unresolved case is part of the residual high-divisibility analysis for generalized Petersen graphs. The theorem records that the existence of χ_{n,2}(G(m,j)) is not known under precisely these conditions.