The Zero Forcing Number of P(n,3) Is Eight for Every n At Least Thirteen

Arnav Krishnan, A correction to the Zero Forcing Number of the Generalized Petersen Graphs P(n,3), arXiv:2607.19412v1, Conjecture 5:

Z(P(n,3)) = 8 for every n ≥ 13.

The paper proves the upper bound for n ≥ 9 (Proposition 4) and the lower bound for 13 ≤ n ≤ 20 by exhaustive search, and states that a lower-bound proof valid for all n is missing.

The frozen theorem D5/S3/Combinatorics/GeneralizedPetersen/ZeroForcingThree.result proves the conjecture for every n ≥ 13. The lower-bound argument transfers a statement about forcing sets into a vertex-isoperimetric inequality for P(n,3), and the isoperimetric inequality is reduced from arbitrary n to finitely many column patterns by deleting empty columns.

Issue 9685 records the screen made before the work: the arXiv record has only v1, the searched formal-conjecture catalogues have no entry, and nothing under Problems/, D5/ or Library/, and no entry of the three screening records, concerns this paper.

1. Upper bound: the eight consecutive outer vertices u_0, …, u_7 force P(n,3) for every n ≥ 9.
2. First forcers: for any finite graph, if a zero forcing set S performs at least p forces, the set X of its first p forcing vertices satisfies |N(X)| ≤ |S|.
3. Isoperimetry: for every n ≥ 14, every 10-vertex set X of P(n,3) has |N(X)| ≥ 8. Seven consecutive empty columns can be shortened by one column without changing |X| or |N(X)|, so it suffices to treat bounded gap patterns; these are exhausted by kernel-checked interval covers of the remaining layer codes.
4. Steps 2 and 3 exclude 7-vertex forcing sets for n ≥ 14.
5. n = 13: explicit forts, checked in the kernel, meet every 7-vertex candidate up to rotation.

The statement would fail if some P(n,3) with n ≥ 13 had a 7-vertex zero forcing set, equivalently if a 10-vertex set of P(n,3), n ≥ 14, had external boundary at most 7 and could be the first-forcer set of such a forcing process, or if a 7-vertex set of P(13,3) met no fort.

SAT checks with CaDiCaL: "|X| = p and |N(X)| ≤ 7" is unsatisfiable for p = 10 and every 14 ≤ n ≤ 40 and satisfiable at n = 13; for p ≤ 9 it is satisfiable for every tested n, so ten first forcers are the least number that works.

theorem; Tier 1 open problem stated in a 2026 paper, preregistered in issue 9685 before the work. The finite parts (the n = 13 fort certificates and the interval covers of bounded gap patterns) are checker instances consumed by the general proof, recorded in the module headers.

The literature screen is bounded to the sources named above; no worldwide priority claim is made.