PreprintCC BY 4.0arXiv
A quadratic refinement of Jackson's \CE\ condition for Hamilton cycles in digraphs
Abstract
For a digraph D, let α2(D) be the largest size of a vertex set no two of whose vertices lie in a common directed $2$-cycle. Let f2(a) be the least integer K such that every K-connected digraph D with α2(D)≤ a has a Hamilton cycle. Jackson proved in 1987 that f2(a)≤2a(a+2)!, whereas a conjecture of Jackson and Ordaz predicts f2(a)≤ a+1. We prove the quadratic bound f2(a)≤12000a2. We also prove that κ(H)≥6000(α(H)+r) guarantees vertex-disjoint paths joining any prescribed r pairs of distinct vertices and covering V(H).
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BibTeX
@misc{ai2026quadratic,
title = {{A quadratic refinement of Jackson's \CE\ condition for Hamilton cycles in digraphs}},
author = {Jiangdong Ai and Yongtang Shi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.08401},
eprint = {2606.08401},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.08401},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A quadratic refinement of Jackson's \CE\ condition for Hamilton cycles in digraphs
AU - Jiangdong Ai
AU - Yongtang Shi
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2606.08401
UR - https://arxiv.org/abs/2606.08401
AB - For a digraph D, let α_2(D) be the largest size of a vertex set no two of whose vertices lie in a common directed $2$-cycle. Let f_2(a) be the least integer K such that every K-connected digraph D with α_2(D)≤ a has a Hamilton cycle. Jackson proved in 1987 that f_2(a)≤2^a(a+2)!, whereas a conjecture of Jackson and Ordaz predicts f_2(a)≤ a+1. We prove the quadratic bound f_2(a)≤12000a^2. We also prove that κ(H)≥6000(α(H)+r) guarantees vertex-disjoint paths joining any prescribed r pairs of distinct vertices and covering V(H).
ER -