Year: 2026
Type: preprint
Abstract: For a digraph $D$, let $α_2(D)$ be the largest size of a vertex set containing no directed $2$-cycle. Let $f_2(a)$ be the least positive integer $k$ such that every $k$-strongly connected digraph $D$ with $α_2(D)\le a$ has a Hamilton cycle. Jackson and Ordaz conjectured that $f_2(a)\le a+1$. Towards this conjecture, we establish the near-linear bound $f_2(a)=O\!\left(\frac{a(\log a)^4}{(\log\log a)^2}\right),$ and hence $f_2(a)=O_\varepsilon(a^{1+\varepsilon})$ for every fixed $\varepsilon>0$. We also disprove the pancyclicity conjecture of Jackson and Ordaz that every digraph $D$ with $κ(D)\geα_2(D)+1$ contains a directed cycle of every length from $2$ to $|V(D)|$.
Source: arXiv (Cornell University)
Authors: Li, Chengli, Ning, Bo
Institutions: South China Normal University, Nankai University
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7220932145