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A correction to the Zero Forcing Number of the Generalized Petersen Graphs P(n,3)

  • Arnav Krishnan
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Abstract

Rashidi, Shajareh Poursalavati, and Tavakkoli [J. Algebra Comb. Discrete Struct. Appl. 7 (2020), no.2, 183-193, Theorem 3.6] claim Z(P(n,3)) = 8 for all n ≥ 12, but this fails at n = 12, where Z(P(12,3)) = 7. We give an explicit $7$-vertex zero forcing set for P(12,3) and its forcing sequence, and verify by exhaustive search that no $6$-vertex set forces P(12,3). We prove Z(P(n,3)) ≤ 8 for n ≥ 9 using an explicit forcing set and rotational symmetry. Exhaustive search yields $Z(P(n,3))$ for 7 ≤ n ≤ 20 and identifies the gap in the published proof: its case analysis fails to exclude $7$-vertex sets. We conjecture Z(P(n,3)) = 8 for n ≥ 13, requiring a matching lower bound for all large n.

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Cite

BibTeXRIS
BibTeX
@misc{krishnan2026correction,
  title = {{A correction to the Zero Forcing Number of the Generalized Petersen Graphs $P(n,3)$}},
  author = {Arnav Krishnan},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2607.19412},
  eprint = {2607.19412},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.19412},
}
RIS
TY  - UNPB
TI  - A correction to the Zero Forcing Number of the Generalized Petersen Graphs P(n,3)
AU  - Arnav Krishnan
PY  - 2026
DA  - 2026/10/08
PB  - arXiv
DO  - 10.48550/arxiv.2607.19412
UR  - https://arxiv.org/abs/2607.19412
AB  - Rashidi, Shajareh Poursalavati, and Tavakkoli [J. Algebra Comb. Discrete Struct. Appl. 7 (2020), no.2, 183-193, Theorem 3.6] claim Z(P(n,3)) = 8 for all n ≥ 12, but this fails at n = 12, where Z(P(12,3)) = 7. We give an explicit $7$-vertex zero forcing set for P(12,3) and its forcing sequence, and verify by exhaustive search that no $6$-vertex set forces P(12,3). We prove Z(P(n,3)) ≤ 8 for n ≥ 9 using an explicit forcing set and rotational symmetry. Exhaustive search yields $Z(P(n,3))$ for 7 ≤ n ≤ 20 and identifies the gap in the published proof: its case analysis fails to exclude $7$-vertex sets. We conjecture Z(P(n,3)) = 8 for n ≥ 13, requiring a matching lower bound for all large n.
ER  -