Skip to content
DraconiticResearch
Search Sources API
203,234 studies31 sourcesUpdated 19 min ago
PreprintarXiv licencearXiv

Hamilton-connected cores and five cycle--wheel Ramsey numbers

  • Zehui Shao
  • Hanxin Jiang
Open at sourcePDFDOICite

Abstract

Let Ws=K1+Cs−1 denote the wheel on s vertices. We give structural proofs that R(C14,W11)=27 and R(C15,W11)=29. Together with the theorem of Chen et al. for n≥16, these equalities give R(Cn,W11)=2n−1 for every n≥14. The two boundary values were included in an earlier survey announcement. We also give structural proofs of R(C8,W7)=15, R(C9,W7)=17, and R(C8,W9)=15. The common starting point is a Hamilton-connected core lemma. For the eleven-vertex wheel, bounds on vertex connectivity and on the matching number of a bipartite graph associated with a local cycle yield a vertex cut of order nine. Paths with prescribed endpoints then rule out every possible pair of orders of the two remaining vertex sets. For the smaller wheels, we use the structure of critical cycle colorings and local cycle-shortening arguments. We also give complete structural classifications of the (C8,C6)- and (C9,C6)-critical colorings, recovering the previously reported counts 24 and 26. All proofs are combinatorial and use no exhaustive graph enumeration.

Related

  1. On Hamilton cycles in connected vertex-transitive graphs of order 2pq
    Zhou Tianlei
    arXivPreprintarXiv licenceFull textCombinatorics & discrete mathematicsarXiv
  2. An explicit connected-sum family with unbounded knot Ramsey-stick gap
    Meng Ji
    arXivPreprintCC BY-NC-SA 4.0Full textGeometry & topologyarXiv
  3. Off-diagonal Ramsey numbers
    Domagoj Bradač
    arXiv (Cornell University)PreprintLicence not statedFull text19 citedCombinatorics & discrete mathematicsOpenAlex
  4. Ramsey numbers R(K1,4,Km,n) for small m
    Khaidar Abdullin, Dmitrii Taletskii
    arXivPreprintCC BY-NC-ND 4.0Full textCombinatorics & discrete mathematicsarXiv
  5. New results on cyclic and ordered Ramsey numbers
    Gaurav Kucheriya, Jan Petr, Jun Yan
    arXivPreprintCC BY 4.0Full textCombinatorics & discrete mathematicsarXiv

Cite

BibTeXRIS
BibTeX
@misc{shao2026hamiltonconnected,
  title = {{Hamilton-connected cores and five cycle--wheel Ramsey numbers}},
  author = {Zehui Shao and Hanxin Jiang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.39508},
  eprint = {2609.39508},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.39508},
}
RIS
TY  - UNPB
TI  - Hamilton-connected cores and five cycle--wheel Ramsey numbers
AU  - Zehui Shao
AU  - Hanxin Jiang
PY  - 2026
DA  - 2026/09/30
PB  - arXiv
DO  - 10.48550/arxiv.2609.39508
UR  - https://arxiv.org/abs/2609.39508
AB  - Let W_s=K_1+C_s−1 denote the wheel on s vertices. We give structural proofs that R(C_14,W_11)=27 and R(C_15,W_11)=29. Together with the theorem of Chen et al. for n≥16, these equalities give R(C_n,W_11)=2n−1 for every n≥14. The two boundary values were included in an earlier survey announcement. We also give structural proofs of R(C_8,W_7)=15, R(C_9,W_7)=17, and R(C_8,W_9)=15. The common starting point is a Hamilton-connected core lemma. For the eleven-vertex wheel, bounds on vertex connectivity and on the matching number of a bipartite graph associated with a local cycle yield a vertex cut of order nine. Paths with prescribed endpoints then rule out every possible pair of orders of the two remaining vertex sets. For the smaller wheels, we use the structure of critical cycle colorings and local cycle-shortening arguments. We also give complete structural classifications of the (C_8,C_6)- and (C_9,C_6)-critical colorings, recovering the previously reported counts 24 and 26. All proofs are combinatorial and use no exhaustive graph enumeration.
ER  -