Skip to content
DraconiticResearch
Search Sources API
203,234 studies31 sourcesUpdated 19 min ago
PreprintCC BY-NC-SA 4.0arXiv

Grundy Total Domination and Skew Zero Forcing in Cartesian Products of Paths and Cycles

  • Fei-Huang Chang
Open at sourcePDFDOICite

Abstract

We determine the Grundy total domination number and the skew zero forcing number for every Cartesian product of two paths or cycles, thereby closing the nonmatching bounds previously known for these rectangular, cylindrical, and toroidal families. For 2≤ a≤ b, Z-(Pa□ Pb) =a−1{a odd, b even}. For p≥2 and c≥3, Z-(Pp□ Cc)= min{p,c},&c is odd, min{2p,c},&c is even. Finally, for 3≤ a≤ b, Z-(Ca□ Cb)= 2a−1,&a=b odd, 2a,&a=b even, min{b,2a},&a<b, a odd, a,&a<b, a even, b odd, 2a,&a<b, a,b even. In every case, γgrt(G)=|V(G)|−Z-(G). The path-containing lower bounds use Kronecker differences of skew-symmetric or hollow symmetric factor matrices, and they also determine maximum skew nullity and minimum skew rank. They further produce an infinite family for which maximum skew nullity is strictly smaller than skew zero forcing. The new cycle--cycle lower bounds use a cyclic column-defect estimate, applied to complete columns when the shorter cycle is odd and, after a one-sided bipartite reduction, to half-columns when both cycles are even. Explicit forcing constructions give matching upper bounds throughout.

Related

  1. On k-limited domination: complexity and Cartesian products
    Aleksandra Tepeh
    arXivPreprintCC BY 4.0Full textCombinatorics & discrete mathematicsarXiv
  2. Sn-equivariant L-classes of Cartesian powers of Witt spaces and the Hodge index conjecture for symmetric products
    Mohammadali Aligholi
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryGeometry & topologyarXiv
  3. The Pairing-Hamiltonian property in Cartesian products of graphs
    Federico Romaniello
    arXivPreprintCC BY 4.0Full textCombinatorics & discrete mathematicsarXiv
  4. Coded Computing for Dynamic System via Cartesian Products
    Chenglin Li, Minquan Cheng, Youlong Wu
    arXivPreprintarXiv licenceFull textGeneral mathematicsarXiv
  5. Well-Covered Powers: Cartesian Products and Cayley Graphs
    Haobo Ma, Thanh Vu, Wenlin Zhang
    arXivPreprintCC BY 4.0Full textCombinatorics & discrete mathematicsarXiv

Cite

BibTeXRIS
BibTeX
@misc{chang2026grundy,
  title = {{Grundy Total Domination and Skew Zero Forcing in Cartesian Products of Paths and Cycles}},
  author = {Fei-Huang Chang},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.27804},
  eprint = {2608.27804},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.27804},
  note = {CC BY-NC-SA 4.0},
}
RIS
TY  - UNPB
TI  - Grundy Total Domination and Skew Zero Forcing in Cartesian Products of Paths and Cycles
AU  - Fei-Huang Chang
PY  - 2026
DA  - 2026/08/28
PB  - arXiv
DO  - 10.48550/arxiv.2608.27804
UR  - https://arxiv.org/abs/2608.27804
AB  - We determine the Grundy total domination number and the skew zero forcing number for every Cartesian product of two paths or cycles, thereby closing the nonmatching bounds previously known for these rectangular, cylindrical, and toroidal families. For 2≤ a≤ b, Z_-(P_a□ P_b) =a−1_{a odd, b even}. For p≥2 and c≥3, Z_-(P_p□ C_c)= min{p,c},&c is odd, min{2p,c},&c is even. Finally, for 3≤ a≤ b, Z_-(C_a□ C_b)= 2a−1,&a=b odd, 2a,&a=b even, min{b,2a},&a<b, a odd, a,&a<b, a even, b odd, 2a,&a<b, a,b even. In every case, γ_gr^t(G)=|V(G)|−Z_-(G). The path-containing lower bounds use Kronecker differences of skew-symmetric or hollow symmetric factor matrices, and they also determine maximum skew nullity and minimum skew rank. They further produce an infinite family for which maximum skew nullity is strictly smaller than skew zero forcing. The new cycle--cycle lower bounds use a cyclic column-defect estimate, applied to complete columns when the shorter cycle is odd and, after a one-sided bipartite reduction, to half-columns when both cycles are even. Explicit forcing constructions give matching upper bounds throughout.
ER  -