Sharp zero forcing numbers for the generalized Petersen graphs P(n,3) and P(n,4)
Abstract
We prove that Z(P(n,3))=8 for every n≥13 and Z(P(n,4))=10 for every n≥18, with both stabilization thresholds sharp, and give the exact values for all smaller admissible n. The first equality proves Krishnan's Conjecture 5; his Proposition 4 supplies its upper bound. The lower bounds use the external boundary of the first forcing sources: ten vertices have boundary at least eight in P(n,3) for n≥14, and sixteen vertices have boundary at least ten in P(n,4) for n≥20. Deletion within an empty column interval reduces these assertions to finite checks, and forts settle the remaining cases. The k=3 stabilization theorem has a Lean 4 formalization; the k=4 proof uses 89 boundary formulas and 11 fort formulas with separately checked DRAT certificates, together with explicit forcing sequences.
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BibTeX
@misc{krishnan2026sharp,
title = {{Sharp zero forcing numbers for the generalized Petersen graphs $P(n,3)$ and $P(n,4)$}},
author = {Arnav Krishnan and Haobo Ma and Wenlin Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.24059},
eprint = {2610.24059},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.24059},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Sharp zero forcing numbers for the generalized Petersen graphs P(n,3) and P(n,4)
AU - Arnav Krishnan
AU - Haobo Ma
AU - Wenlin Zhang
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.24059
UR - https://arxiv.org/abs/2610.24059
AB - We prove that Z(P(n,3))=8 for every n≥13 and Z(P(n,4))=10 for every n≥18, with both stabilization thresholds sharp, and give the exact values for all smaller admissible n. The first equality proves Krishnan's Conjecture 5; his Proposition 4 supplies its upper bound. The lower bounds use the external boundary of the first forcing sources: ten vertices have boundary at least eight in P(n,3) for n≥14, and sixteen vertices have boundary at least ten in P(n,4) for n≥20. Deletion within an empty column interval reduces these assertions to finite checks, and forts settle the remaining cases. The k=3 stabilization theorem has a Lean 4 formalization; the k=4 proof uses 89 boundary formulas and 11 fort formulas with separately checked DRAT certificates, together with explicit forcing sequences.
ER -