Existence of remainder-one colorings in unresolved generalized Petersen cases

Determine whether a closed coloring with remainder 1 modulo n exists for the generalized Petersen graph G(m,j) in the cases where 8 divides n, 2 divides j, and either 16 does not divide n while 4 divides m, or 8 divides m.

Background

The paper gives a near-complete divisibility-based classification of the existence of χ_{n,1}(G(m,j)) for generalized Petersen graphs. The unresolved regimes occur when n is divisible by 8, j is even, and m is divisible by 4; the authors distinguish the subcases 8|n, 16∤n, 4|m and 8|n, 8|m.

The preceding arguments establish nonexistence in several neighboring high-2-divisibility cases and show that the labeling scheme used in the paper fails in the unresolved regimes. The authors explicitly leave open the possibility that more exotic labelings may establish existence.

References

If $8\mid n$, $16\nmid n$, $4\mid m$, and $2\mid j$, existence of $\chi_{n,1}(G(m,j))$ is not currently known. If $8\mid n$, $8\mid m$, and $2\mid j$, existence of $\chi_{n,1}(G(m,j))$ is not currently known.

— Chromatic numbers with closed local modular constraints  (2503.00406 - Herden et al., 1 Mar 2025) in Theorem 7.1, items 7–8, and the discussion following Lemma 7.5 in Section 7 (Generalized Petersen Graphs)