A Sublinear Minimum-Degree Condition for $2$-Connected Subgraphs of All Orders

@misc{ozeki2026sublinear,
  title = {{A Sublinear Minimum-Degree Condition for $2$-Connected Subgraphs of All Orders}},
  author = {Kenta Ozeki and Takahiro Ueoro},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.17478},
  eprint = {2608.17478},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.17478},
}

TY  - UNPB
TI  - A Sublinear Minimum-Degree Condition for $2$-Connected Subgraphs of All Orders
AU  - Kenta Ozeki
AU  - Takahiro Ueoro
PY  - 2026
DA  - 2026/08/30
PB  - arXiv
DO  - 10.48550/arxiv.2608.17478
UR  - https://arxiv.org/abs/2608.17478
AB  - Motivated by an analogue of pancyclicity, we study minimum-degree conditions ensuring that a $2$-connected graph G of order n contains a $2$-connected subgraph of every order ℓ∈{4,5,…,n}. Yin and Wu [A minimum degree condition for a 2-connected graph containing all possible orders of 2-connected subgraphs, Discrete Appl. Math. 387 (2026), 129-136] initiated the study of this problem and showed that the condition δ(G)≥ ⌈ n/3⌉+1 is sufficient. Kashima conjectured that the condition δ(G)≥ √(3n) is sufficient. In this paper, we prove that every $2$-connected graph G of order n with δ(G)≥ 2n^2/3+6n^1/3+2 contains a $2$-connected subgraph of every order from $4$ to n. In particular, this gives the first sufficient minimum-degree condition of sublinear order in n.
ER  -