bibkey: krishnan2026correction authors: Arnav Krishnan year: 2026 title: “A correction to the Zero Forcing Number of the Generalized Petersen Graphs P(n,3)” doi: 10.48550/arXiv.2607.19412 url: https://arxiv.org/abs/2607.19412v1 claim: “Conjecture 5. Z(P(n,3)) = 8 for every n ≥ 13.” strata_touched:

• D5/S3/Combinatorics/GeneralizedPetersen/ZeroForcingThree license: citation-only triage: anchor

Krishnan, the zero forcing number of P(n,3)

The note corrects a published claim that Z(P(n,3)) = 8 for all n ≥ 12: it exhibits a 7-vertex zero forcing set of P(12,3), proves the upper bound Z(P(n,3)) ≤ 8 for n ≥ 9, computes Z(P(n,3)) for 7 ≤ n ≤ 20 by exhaustive search, and states the corrected lower bound as a conjecture.

Verified locator

DOI: 10.48550/arXiv.2607.19412

URL: https://arxiv.org/abs/2607.19412v1

The arXiv record shows only v1 (13 July 2026).

• Locator: Proposition 4, “For every n ≥ 9, S = {u_0, …, u_7} is a zero forcing set of P(n,3).”
• Locator: Table 1, Z(P(n,3)) for 7 ≤ n ≤ 20, equal to 8 for 13 ≤ n ≤ 20 and to 7 for n = 11, 12.
• Locator: Conjecture 5, “Z(P(n,3)) = 8 for every n ≥ 13.” The following paragraph reads: “the content is the lower bound Z(P(n,3)) ≥ 8 for all n ≥ 13, which we verified only for n ≤ 20. Proving it for all n means excluding 7-vertex forcing sets uniformly in n”.
• Locator: Section 5, “Whether they stay 8 for all larger n (Conjecture 5) is open”.

Reading of the statement

P(n,3) has outer vertices u_i, inner vertices v_i (indices mod n), outer-cycle edges u_i u_{i+1}, spokes u_i v_i and inner edges v_i v_{i+3}. A set S is zero forcing when repeatedly letting a black vertex with exactly one white neighbour colour that neighbour black turns every vertex black; Z is the least size of such a set. Conjecture 5 asserts Z(P(n,3)) = 8 for every n ≥ 13.

Scope of the recorded answer

The conjecture holds. The upper bound is Proposition 4. For the lower bound, if a zero forcing set performs at least p forces then the set X of its first p forcing vertices satisfies |N(X)| ≤ |S|, where N(X) is the external vertex boundary; for n ≥ 14 every 10-vertex set of P(n,3) has |N(X)| ≥ 8, which excludes 7-vertex forcing sets, and n = 13 is settled by explicit forts meeting every 7-vertex candidate.

Bounded prior-resolution evidence

Read on 2026-09-24 and 2026-09-26: the arXiv record (v1 only); google-deepmind/formal-conjectures, conjectures.io and mathdb by title, arXiv number and the terms “zero forcing” and “Petersen” returned no match. This is a bounded negative finding.