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PreprintarXiv licencearXiv

An Exact Dominant Degree Condition for Transitive Tournament Factors in Digraphs

  • Yufei Chang
  • Shuo Wei
  • Jin Yan
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Abstract

Let r≥2, let Tr denote the transitive tournament on r vertices, and write dG*(v):=max{dG+(v),dG-(v)}. We prove that if r| n and an n-vertex digraph G satisfies dG*(x)+dG*(y)≥ 2(1−1/r)n−1 for every x≠ y ∈ V(G) with xy ∉ E(G), then G has a Tr-factor, and the bound is best possible. Furthermore, by applying our main theorem, we settle Treglown's conjecture on the dominant degree d*G(x) ≥ (1−1/r)n and answer Molla and Treglown's problem of determining the exact Ore-type threshold $2(1-1/r)n - 1$, and we obtain stronger versions of the theorems of Czygrinow, DeBiasio, Kierstead and Molla.

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Cite

BibTeXRIS
BibTeX
@misc{chang2026exact,
  title = {{An Exact Dominant Degree Condition for Transitive Tournament Factors in Digraphs}},
  author = {Yufei Chang and Shuo Wei and Jin Yan},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2608.10445},
  eprint = {2608.10445},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.10445},
}
RIS
TY  - UNPB
TI  - An Exact Dominant Degree Condition for Transitive Tournament Factors in Digraphs
AU  - Yufei Chang
AU  - Shuo Wei
AU  - Jin Yan
PY  - 2026
DA  - 2026/09/08
PB  - arXiv
DO  - 10.48550/arxiv.2608.10445
UR  - https://arxiv.org/abs/2608.10445
AB  - Let r≥2, let T_r denote the transitive tournament on r vertices, and write d_G^*(v):=max{d_G^+(v),d_G^-(v)}. We prove that if r| n and an n-vertex digraph G satisfies d_G^*(x)+d_G^*(y)≥ 2(1−1/r)n−1 for every x≠ y ∈ V(G) with xy ∉ E(G), then G has a T_r-factor, and the bound is best possible. Furthermore, by applying our main theorem, we settle Treglown's conjecture on the dominant degree d^*_G(x) ≥ (1−1/r)n and answer Molla and Treglown's problem of determining the exact Ore-type threshold $2(1-1/r)n - 1$, and we obtain stronger versions of the theorems of Czygrinow, DeBiasio, Kierstead and Molla.
ER  -