Runs of consecutive outer vertices in generalized Petersen graphs, and two erroneous published values of the zero forcing number
Abstract
Let GP(n, k) be the generalized Petersen graph and Z the zero forcing number. The standard upper bound Z(GP(n, k)) ≤ 2k+2 is witnessed by a run of 2k+2 consecutive vertices of the outer cycle. We prove that this witness cannot be shortened inside its own shape: for every k ≥ 1, every n ≥ max(2k+3, 5k-1) and every starting index j, the shortest run of consecutive outer vertices that is a zero forcing set of GP(n, k) has length exactly 2k+2. The lower half of that statement is proved by exhibiting, for each k, an explicit set of 6k-2 vertices — a number independent of n — which contains the run of length 2k+1, misses the next outer vertex, and is closed under the colour-change rule. At k=3 this turns the negative half of a computation Krishnan reports for 10 ≤ n ≤ 300 — that seven consecutive outer vertices do not force — into a theorem for every n ≥ 13, and gives the first statement we are aware of that rules out a family of candidate 7-element sets uniformly in n, which is the shape of the open half of his Conjecture 5. Alongside these we record a lower bound Z(GP(n, k)) ≥ 4 valid for all n with 2k<n and n ≠ 3k, the resulting exact value Z(GP(n, 1))=4 for all n ≥ 5, and two published values of Z that are wrong: the claim Z(GP(n, 3))=8 for n ≥ 12 of Rashidi, Shajareh Poursalavati and Tavakkoli, whose failure at n=12 was found by Krishnan and is confirmed here, and the entry Z(GP(7, 2))=5 printed in the ℓ=0 column of Table 1 of arXiv:2508.02564v1, whose true value is 6. Every theorem here is machine-checked in Lean 4, and each one quantified over all n is proved without computation.
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Archived files
- Version 1 · current (opens in a new tab)
Source snapshot 2026-08-30 15:34 UTC
File fingerprint
cf9db86f587536c46cfd759ce64a18f9f30b7cebe6a30523833eb042b07930e4
Claim ledger
Stated results
ZF1routine2026-08-23
The colour-change rule, its closure, monotonicity, and termination – the reasoning layer, with no data in it
ZF2routine2026-08-23
The bitmask engine computes the colour-change rule – proved, not assumed
ZF3known2026-08-23
Five published Z(P(n,k)) values, reproduced before anything new is stated
ZF4known data2026-08-23
Z(P(n,3)) for 7 <= n <= 13: 6, 6, 6, 8, 7, 7, 8 – every non-repeating cell of the source's Table 1
ZF5known2026-08-23
THE ADJUDICATION: [RPT2020, Theorem 3.6] – 'Z(P(n,3)) = 8 for all n >= 12' – is FALSE, and Krishnan's correction is confirmed
ZF6known data2026-08-23
The conjectured tail, n = 14 and 15: still 8
ZF7routine2026-08-23
Negative controls: the graph is the graph, the rule is non-monotone at one vertex, the search is not vacuous, and the bound is refutable in both directions
ZF8routine2026-08-28
Rotation equivariance of the colour-change rule: ρ: uᵢ ↦ uᵢ₊₁, vᵢ ↦ vᵢ₊₁ is an automorphism of P(n,k) and the closure commutes with it
ZF9known2026-08-28
Z(P(n,k)) ≤ 2k+2 for every k ≥ 1 and every n ≥ 2k+3, by 2k+2 consecutive outer vertices — with the threshold the published proofs do not state
ZF10known2026-08-28
arXiv:2607.19412 Proposition 3.1 — Z(P(n,3)) ≤ 8 for all n ≥ 9 — as a kernel theorem, the family's first statement about any n outside the searched range
ZF11candidate2026-08-28
Seven consecutive outer vertices force P(n,3) for no n ≥ 13 — the source's 10 ≤ n ≤ 300 computational claim, for all n, and the first uniform-in-n piece of the open Conjecture 4.2
ZF12routine2026-08-28
Negative controls for the all-n layer: the rotation is the automorphism, the cascade is exactly three rounds, the stall set is exactly the closure, and the bound is refutable in both directions
ZF13candidate2026-08-28
The general-k stall: 2k+1 consecutive outer vertices force P(n,k) for no k ≥ 1 and no n ≥ 5k-1
ZF14candidate2026-08-28
Where a consecutive outer run starts does not matter, and 2k+2 is exactly the least forcing run length
ZF15routine2026-08-28
Negative controls for the general-k and orbit layers
ZF16known data2026-08-28
Z(P(n,2)) for 6 ≤ n ≤ 9: 4, 6, 5, 6
ZF17candidate2026-08-28
THE SECOND ADJUDICATION: arXiv:2508.02564v1 Table 1's ℓ = 0 column is false at Z(P(7,2))
ZF18routine2026-08-28
Z(P(n,k)) ≥ 4 for every k ≥ 1 and every n with 2k < n, n ≠ 3k — the family's first uniform-in-n lower bound
ZF19known2026-08-28
Z(P(n,1)) = 4 for every n ≥ 5 — the family's first exact value of Z valid for all n, kernel-clean
Provenance
- Generated by
- Machina Mathematica
- Released by
- Korea Superintelligence Labs
- Source context
- Z(P(n,3)), the zero forcing number of the generalized Petersen graph — and a published theorem that turns out to be false at the first case in its own range.
- Snapshot
- 2026-09-07 03:53 UTC
- Ledger commit
801848d7